The path of a fast moving particle traces a circle with equation (x-3)² + (y-4)² = 25. It starts at point (3,-1), moves counter clockwise, and passes the point (8,4) for the first time after traveling 7 microseconds. Where is the particle after traveling for 20 microseconds.
Answers
The particle is at the point (-1.112,3.909) after traveling for 20 microseconds.
We have been given a circle,
(x-3)²+(y-4)²=25
whose center is (3,4)
and with a radius of 5 units.
The particle starts at (3,-1) which is the 6 o'clock position in the circle, and ends at (8,4) after 7 microseconds which are at the 3 o'clock position in the circle.
So, the particle traveled 1/4 of the circumference of the circle in 7 microseconds
therefore according to the unitary method it will travel 5/7 of the circumference of the given circle.
so it will subtend 2π×5/7 of the circle from (3,-1)
from the polar form of the circle and the shifting of center of the circle we can conclude
that it's location will be
(5 × cos (10π/7) - 3 , 5 × sin (10π/7) - 4)
= (-1.112,3.909)
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Related Questions
$4700 accumulating to $5994.76, compounded monthly for 5 years. What is the interest rate %
Answers
48.76%
Step-by-step explanation:Compound interest describes interest on the principal plus interest.
Compound Interest Formula
The formula to find compound interest is [tex]A=P(1+\frac{r}{n})^{nt}[/tex]. In the formula:
A is the total amount, P is the principal, r is the rate as a decimal, n is how often interest is compounded,t is time.We can plug in our information to find the rate.
Solving For Interest Rate
First, let's rewrite the equation with our information. Note that for simplicity I will write rounded values for each step, but in reality, no rounding should be done until the last step.
[tex]5994.76 = 4700(1+\frac{r}{12})^{12*5}[/tex]Then, simplify the exponent and divide both sides by 4700.
[tex]1.275=(1+\frac{r}{12} )^{60}[/tex]Take the 60th root of both sides.
[tex]1.004=(1+\frac{r}{12} )[/tex]Subtract 1 from both sides.
[tex]0.004063=\frac{r}{12}[/tex]Finally, multiply both sides by 12.
r = 0.04876This means that the rate is 0.04876. We can multiply this by 100 to get the rate as a percentage. The interest rate is 48.76%.
Prove the Pythagorean Theorem with Similar Triangles using a two-column proof.
Answers
Answer:
u
Step-by-step explanation
Company A has 18 machines that each produce t toy cars per hour. Company B has 15 machines that each produce 75 more toy cars per hour than each of Company A’s machines. In order for the companies to produce an equal number of toy cars, how many toy cars does each machine at Company A need to produce per hour?
Write an equation that can be used to solve the problem.
Part B
Feedback
Incorrect
2 tries left. Please try again.
Fill in the blank question.
Company A’s machines each produce
toy cars per hour.
Company B’s machines each produce
toy cars per hour.
Answers
The system of equations is solved and the number of toy cars of each machine of Company A should produce to have equal number of toy cars is t = 375 toys
What is an Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the number of toy cars produced by Company A be A
Let the number of toy cars produced by Company B be B
The number of machines for A = 18 machines
The number of machines for B = 15 machines
The number of toy cars produced by A = t cars
The number of toy cars produced by B = 75 + t
And ,
The total number of toy cars of A = 18 x t
The total number of toy cars of B = 15 ( 75 + t )
So , in order for both the company's to have the same toys is
18t = 15 ( 75 + t )
On simplifying the equation , we get
18t = 1125 + 15t
Subtracting 15t on both sides of the equation , we get
3t = 1125
Divide by 3 on both sides of the equation , we get
t = 375 toys
Hence , the equations is A = 18t and B = 15 ( 75 + t )
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Select True or False for each statement. True False The equation y=3x−5 represents a function. True – The equation , y equals 3 x minus 5, represents a function. False – The equation , y equals 3 x minus 5, represents a function. A function can only be represented by a straight line on the coordinate plane. True – A function can only be represented by a straight line on the coordinate plane. False – A function can only be represented by a straight line on the coordinate plane.
Answers
The statement "The equation y = 3x - 5 represents a function." is; TRUE
The statement "A function can only be represented by a straight line on the coordinate plane." is; FALSE
How to represent Linear Functions?A linear function is one that is formed when a variable y varies directly with a variable x. That is, when y increases, x also increases and the opposite is also true.
The equation y = 3x - 5 represents a function. This statement is said to be true because every value of x gives a different value of y and as such we can say it maps different elements to different images.
A function can only be represented by a straight line on the coordinate plane. This statement is false because a function can also be represented by a circle, ellipse, parabola, etc.
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The constant of proportioality between the number of children (c) on a field trip and the number of teachers (t) on the trip is 14/3
Answers
The constant of proportionality is known to be the ratio of two proportional values that is said to be in a constant value.
What is constant of proportionality?This means that for every 3 teachers present on the trip, there will be 14 children. This proportion can be represented by the equation c= (14/3)t, where c represents the number of children and t represents the number of teachers. This proportionality can be used to predict the number of children on a trip based on the number of teachers, or vice versa. For example, if there are 9 teachers on a trip, we can use the equation to predict that there will be (14/3) * 9 = 42 children on the trip. This proportionality also implies that as the number of teachers increases, the number of children will also increase in the same proportion. It is represented as a number, often represented by the letter "k", that represents the ratio between the two variables. In a proportion, the two variables are related in a fixed ratio, meaning that if one variable increases, the other variable will also increase in the same proportion. For example, if there is a constant of proportionality of 2 between the number of apples (a) and the number of oranges (o), it means that for every 2 apples there are, there will be 1 orange.To learn more about proportionality refer to:
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A coffee shop sells bags of coffee for
$9.90 per kilogram. Each bag holds 0.5 kilogram of coffee.
How many bags of coffee do they sell if they earn $702.90?
Explain your answer.
Answers
Answer:
142 bags
Step-by-step explanation:
0.5KG is half of a KG, so we can start by finding half of $9.90
$9.90/2 being $4.95
So each bag is $4.95
The total is $702.90
Dividing $702.90/$4.95 will equal the number of bags
702.9/4.95=142 bags
The table shows some ordered pairs that belong to quadratic function h. What is the range of h?
Answers
Answer:
The range of a function is the set of all possible outputs (or y-values) of the function. To find the range of h, we can look at the y-values in the table.
The y-values for h(x) are: -27, -13, -3, 3, 5, 3, -3
The range of h is the set of all these y-values. We can see that the lowest y-value is -27 and the highest y-value is 5. So, the range of h is {-27,-13,-3,3,5}
Therefore, the range of h is from -27 to 5.
Solve for t
18,000=9000(1.003)^12t
Answers
Answer:
t=1.92938456
Step-by-step explanation:
Isolate the variable by dividing each side by factors that don't contain the variable.
2. Write an inequality statement whose solution is an empty set.
3. Write an inequality statement whose solution is some but not all real numbers. Graph the solution on a number line.
4. Write an inequality statement whose solution is all real numbers. Graph the solution on a number line.
5. Write an inequality statement whose solution is exactly one number. Graph the solution on a number line.
Answers
The inequality statements are 1 < 0, x > 0, x = x and x = 5
How to determine the inequality statementsAn empty set
An empty set is a set with no elements.
An inequality statement with no solutions results in an empty set.
One example of an inequality statement with no solutions is 1 < 0.
Some solution but not all real numbers
An inequality statement whose solution is some but not all real numbers is x > 0.
The solution set is the set of all positive real numbers, excluding 0.
The number line is represented as follows
|------------------------------------| 0 |----------------------------- |
All real numbers
A statement whose solution is all real numbers is x = x
The solution set is the set of all real numbers
The number line is represented as follows
|--------------------------- | ... | -2 | -1 | 0 | 1 | 2 | ... |
Exactly one number
A statement whose solution is exactly one number is x = 5.
The solution set is the set containing only the number 5
The number line is represented as follows
|--------------------------- | ... | 4 | ... |
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Label each item as factor by grouping or factor using substitution.
To factor a trinomial of the form
ax 2 + bx + c, find a factor pair of ac
that has the sum of b. Rewrite bx as a
sum of those factors. Then factor out
the GCF from the two groups of terms
to write the original trinomial as the
product of two binomials.
Answers
Answer:
To factor a trinomial of the form ax^2 + bx + c, you can use the following steps:
Step-by-step explanation:
To factor a trinomial of the form ax^2 + bx + c, you can use the following steps:
Find a factor pair of ac that has the sum of b. For example, if a = 2, b = -7, and c = -21, then the factor pair for -42 (2 * -21) that has the sum of -7 is (-7, -6).
Rewrite bx as a sum of those factors. In the example, -7x = -7x + (-6x).
Factor out the GCF (greatest common factor) from the two groups of terms. In the example, (2x - 7)(x - 6) = 2x(x - 6) - 7(x - 6) = 2x^2 - 14x + 12x - 42 = 2x^2 - 2x - 42.
Write the original trinomial as the product of two binomials. In the example, the trinomial 2x^2 + -7x + -21 can be factored as (2x - 7)(x - 6)
So the final factorization is (2x - 7)(x - 6)
16x² + y² - 128x - 20y + 292 = 0
Find the focal radius of the ellipse and the points at which the two foci sit.
Answers
The focal radius of the ellipse of which the two foci seat are:
(-8 + 6√3, 10) (-8 - 6√3, 10).How to solve for the focal radiusAn ellipse can be represented in the standard form as (x/a)² + (y/b)² = 1, where (a,b) is the center of the ellipse and a and b are the semi-major and semi-minor axes respectively.
Given the equation of the ellipse:
16x² + y² - 128x - 20y + 292 = 0
To put it in the standard form, we have to complete the square and then divide both sides by the constant on the right-hand side.
First, we complete the square by adding and subtracting (128/2)² and (20/2)² respectively:
16x² - 128x + (128/2)² + y² - 20y + (20/2)² = 292 + (128/2)² + (20/2)²
Then, we divide both sides by 292:
(16x² - 128x + (128/2)²)/292 + (y² - 20y + (20/2)²)/292 = 1
On simplifying we get:
(4x - 8)²/144 + (y - 10)²/36 = 1
Now we have the standard form of the equation of the ellipse and we can find the semi-major and semi-minor axis and the center of the ellipse.
The semi-major axis is equal to the square root of the coefficient of x squared (144) and the semi-minor axis is equal to the square root of the coefficient of y squared (36).
The center of the ellipse is (-8, 10) and the semi-major and semi-minor axis are 12, 6 respectively.
The focal radius is the distance between the center and the focus. The focal radius is equal to the square root of the semi-major axis squared minus the semi-minor axis squared.
Focal radius = √(a² - b²) = √(144-36) = √108 = 6√3
The two foci of the ellipse sit on the x-axis, symmetric about the center of the ellipse. The foci are located at (-8 ± 6√3, 10).
Therefore, the two foci of the ellipse are at (-8 + 6√3, 10) and (-8 - 6√3, 10).
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if m< is 28 degrees, what is m< DAC
Answers
We can write the measurement of ∠DAC as -
∠DAC = ∠ADB - ∠DCA
What is a triangle?A triangle is a polygon with three edges and three vertices.
Given is a triangle ABC.
The external angle is the sum of two interior opposite angles.
We can write -
∠ADB = ∠DAC + ∠DCA
So, we can write -
∠DAC = ∠ADB - ∠DCA
Therefore, we can write the measurement of ∠DAC as -
∠DAC = ∠ADB - ∠DCA.
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Use your formula to determine the height of a trapezoid with an area of
24 cubic centimeters and base lengths of 9 cm and 7 cm
Formula: 1/2(a+b)h
Answers
The height of a trapezoid is 3cm
How to use the formula for area to determine the height of a trapezoid?The formula for the area of a trapezoid is:
A = 1/2(base1 + base2) × height
Where base1 and base2 are the lengths of the parallel sides of the trapezoid, and height is the distance between the bases.
That is:
A = 1/2(a+b)h
To determine the height of a trapezoid using the formula substituting the given values and solve for h. That is:
A = 1/2(a+b)h
where A = 24 cubic centimeters, a = 9 cm and b = 7cm
24 = 1/2(9+7)h
24 = 1/2(16)h
24 = 8h
8h = 24
h = 24/8
h = 3 cm
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Someone baked 2 types of pie.
He served 1/2 of the blueberry pie. And he also served 1/4 of the Apple pie, If each pie had 8 pieces to start, what fraction in eights of the Apple did he served?
Answers
The fraction in eights of the Apple pie that the person baked, is 2 / 8 of the apple pie.
How to find the fraction ?The number of pieces that the apple pie had was 8 pieces to start. This means that if the baker served 1 / 4 of this, you can find the fractions in eights by multiplying the fraction served by the number of pieces :
= Fraction served x Number of pieces
= 1 / 4 x 8
= 8 / 4
= 2 pieces
The fraction in eights of the Apple that was served was :
= Number of pieces served / Number of total pieces
= 2 / 8
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то 13.) Solve the right triangle. Round answers to the nearest tenth. To receive full credit you must show all work. R S 57° 15 T RT = RS= m/T=
Answers
Answer:
RT = 17.9
RS = 9.7
m<T = 33°
Step-by-step explanation:
sin R = opp/hyp
sin R = ST/RT
sin 57° = 15/RT
RT = 15/sin 57°
RT = 17.9
cos R = adj/hyp
cos R = RS/RT
cos 57° = RS/17.88544939
RS = 17.88544939 × cos 57°
RS = 9.7
m<R + m<S + m<T = 180°
57° + 90° + m<T = 180°
m<T = 33°
How much interest will you pay over the life of a $220,000 30-year loan at 8 percent with monthly payments of $1,614.28?
Answers
The total interest that will be paid over the life is $5,28,000. The solution has been obtained using arithmetic operations.
What are arithmetic operations?
For all the real numbers, there are four basic mathematical operations which are:
1. Addition(‘+’) wherein the sum of the numbers is obtained.
2. Subtraction(‘-’) wherein the difference of the numbers is obtained.
3. Multiplication(‘×’) wherein the product of the numbers is obtained.
4. Division(‘÷’) wherein the quotient of the numbers is obtained.
We are given 30 year loan amounting $220,000 at interest rate of 8 percent with monthly payments of $1,614.28.
The annual interest comes out to be
8% of $220,000 = $17,600
The total interest = $17,600 * 30 = $5,28,000
Hence, the total interest that will be paid over the life is $5,28,000.
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a and b are parallel. Find the missing angles.
Answers
Step-by-step explanation:
....................
Answer:
Step-by-step explanation:
top=125 degrees
second from the top= 55 degrees
3rd=35 degrees
4th=55 degrees
5th=125 degrees
X/4 = 20
please help
Answers
Answer:
x = 80
Step-by-step explanation:
Isolate the variable, x. Note the equal sign, what you do to one side, you do to the other. Note that x is being divided by 4. To isolate x, do the opposite. Multiply 4 to both sides of the equation:
[tex]\frac{x}{4} = 20\\(\frac{x}{4} ) * 4 = (20) * 4\\x = 20 * 4\\x = 80[/tex]
x = 80 is your answer.
~
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In an experiment with a bag of marbles, P(green) = three fourths. Interpret the likelihood of choosing a green marble.
Likely
Unlikely
Equally likely and unlikely
This value is not possible to represent probability of a chance event.
Answers
The probability of picking a green marble is 3/4 or 0.75. This means it is likely the event would occur. Therefore, option A is the correct answer.
What is the probability?Probability can be defined as the ratio of the number of favourable outcomes to the total number of outcomes of an event.
We know that, probability of an event = Number of favourable outcomes/Total number of outcomes
Given that, an experiment with a bag of marbles, P(green)=3/4.
Probability is used to determine how likely it is that a random event would happen. The probability that a random event occurs lie between 0 and 1. The more likely the event is to happen, the closer the probability value would be to 1. The less likely it is for the event not to happen, the closer the probability value would be to zero.
The probability of picking a green marble is 3/4 or 0.75. 0.75 is more than 0.5. This means it is likely the event would occur.
Therefore, option A is the correct answer.
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Let f(x)-3x+5, h(x)-4x-2 and g(x)=x^2. Write an expression for each function.
a. f+g
b. f-g
Answers
Answer:
a. f+g = -3x+5 + x^2
b. f-g = -3x+5 - x^2
Step-by-step explanation:
g(x) = x^2
I'll assume the other two equations were meant to be:
f(x) = -3x+5, and h(x) = -4x-2
a. f+g = -3x+5 + x^2
b. f-g = -3x+5 - x^2
if f(x) = 4x^3+24x^2+41x+15 and x+3 is a factor of f(x) then find all of the zeros of f(x) algebraically
Answers
All of the zeros of f(x) algebraically are; x = -3, x = -1/2, x = -5/2
How to find the zeros of the polynomial?The zeros of a polynomial p(x) are defined as all the x-values that make the polynomial equal to zero.
The polynomial is;
f(x) = 4x³ + 24x² + 41x + 15
To find the x-intercept, we will equate the polynomial to zero to get;
4x³ + 24x² + 41x + 15 = 0
Since (x + 3) is a factor, then we can factorize the polynomial as;
(x + 3)(4x² + 12x + 5) = 0
Rewriting the quadratic term gives;
(x + 3)(4x² + 2x + 10x + 5) = 0
(x + 3)(2x(2x + 1) + 5(2x + 1)) = 0
Thus, the factors are expressed as;
(x + 3)(2x + 1)(2x + 5) = 0
Thus, the zeros are;
x + 3 = 0
x = -3
2x + 1 = 0
x = -1/2
2x + 5 = 0
x = -5/2
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Greta is looking for a new fishing pole for an upcoming trip to the Grand Canyon. She found the model she likes at a local sporting goods store. The price of the pole is $59,50 with a sales tax of 6%. However, she can purchase the pole on a website for $62.50 with no sales tax added on to the price. What is the difference in price between the store and the website? Which is more expensive?
Answers
To find the total cost of the pole at the store including sales tax, we need to multiply the price of the pole by the sales tax rate (expressed as a decimal).
$59.50 x 0.06 = $3.57
So the total cost of the pole at the store is $59.50 + $3.57 = $63.07.
To find the difference in price between the store and the website, we can subtract the website price from the store price:
$63.07 - $62.50 = $0.57
So the pole is $0.57 more expensive at the store than on the website.
A parking lot is 80 feet wide by 120 feet long. If the length and width are increased by the same length, what polynomial represents the area of the new lot? What is the new area of the increase is 25 feet?
Answers
The polynomial that represents the area of the new lot is x² +200x +9600 and the new area of the increase is 25 feet is 15225 ft²
What is Polynomial?A polynomial is an expression made up of variables (also known as indeterminates) and coefficients that only employs the addition, subtraction, multiplication, and non-negative integer exponents of the variables.
A single variable, x, is used as the only variable in the indeterminate polynomial x² - 4x + 7. Polynomials are used in many areas of science and mathematics.
They are employed in calculus and numerical analysis to approximate other functions, as well as in the definition of polynomial functions, which are used in a variety of fields, from basic chemistry and physics to economics and social science.
Simple word problems to sophisticated scientific conundrums can all be represented using polynomial equations.
Polynomials are used to create algebraic varieties in higher mathematics.
Area = Length x Breadth
Area =(80 + x)x(120 + x) = x² +200x + 9600
Area = (80 + 25)x(120 + 25) = 105x145
Area = 15225 ft²
Therefore, the new area of the plot is 15225 ft²
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Please Help Me!!!!! Sorry about the small picture!
Answers
Answer:
1:12, then 1 :11
wait unless its like in the same thing
wait sorry im confused
if its that then i think it would be 2:12
Step-by-step explanation:
sorry not too sure!!!
The equation, 3(3x - 10) = 5(x +10) shows the relationship between the perimeter of an equilateral triangle and the perimeter of a regular pentagon.What is the perimeter of the pentagon?
The equation, 3(3x - 10) = 5(x +10) shows the relationship between the perimeter of an equilateral triangle and the perimeter of a regular pentagon.What is the perimeter of the pentagon?
100
20
50
150
Answers
a certain phone company charges $4.50 for the first five minutes of an international phone call. additional time is charged at $.50 per minute. how much would a customer be charged for an international phone call that started at 9:35 p.m. and ended at 11:15 p.m. the same day?
Answers
A customer would be charged $ 52 for an international phone call that started at 9:35 p.m. and ended at 11:15 p.m. the same day.
Charge for first 5 charged = $ 4.50
Charge for additional time = $ 0.50 per minute
Starting time = 9:35 p.m.
End time = 11:15 p.m.
Total minutes = 100 minutes
Total charge = 4.50 + (95 x 0.50)
= 4.50 + 47.50
= 52.00
Hence, a customer would be charged $ 52 for an international phone call that started at 9:35 p.m. and ended at 11:15 p.m. the same day i.e. for 100 minutes.
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Find the Area of the figure below, composed of a rectangle and one semicircle, with
another semicircle removed. Round to the nearest tenths place.
13
12
Answers
Step-by-step explanation:
that is really simple, because the additional half-circle compensates for the missing half-circle on the other side.
so, the area is plain and simple the area of the rectangle, which is
13×12 = 156 = 156.0
Do you know what 20% of 60 is?
Answers
Answer:1
Step-by-step explanation:
0
in the given figure,AB is parallel to CD and angle EAB =50°,angle ECD =100°,find angle AEC
Answers
If AB is parallel to CD and angle EAB =50°,angle ECD =100° then Angle AEC is 112∘
Since AB∥CD
∴∠CAB+∠DCA=180 ∘ (Co-interior angles)
The inside angles total 180 degrees and are known as co-interior angles. It indicates that two internal angles that are on the same side of the transversal have a sum that is additional.
∴22 ∘ +∠DCA=180 ∘
⇒∠DCA=180 ∘ −22 ∘ =158 ∘
Also, ∠ECD+∠DCA+∠y=360 ∘ (Angles about a point)
Angles around a point refer to the total number of angles that can be combined to produce a complete turn. A point's surrounding angles add up to 360°. The angles circling a point add up to 360° since they have completed a full turn and are identical in magnitude.
⇒90 ∘ +158 ∘ +∠y=360 ∘
⇒∠y=360 ∘ −(90 ∘ +158 ∘ )
⇒∠AEC=y=360 ∘ −248 ∘ =112∘
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Find the equation of the plane passing P(1,2,1) and is orthogonal to the two planes: x-y-z-10 = 0, x-2y + z-2=0.
Answers
The equation of the plane passing P (1,2,1) and orthogonal to the two planes: x-y-z-10 = 0, x-2y + z-2=0 is -3x-2y-z+8=0.
Equation of plane passing through (x1,y1,z1) is given by
A(x-x1)+B(y-y1)+C(z-z1)=0
where, A, B, and C are direction ratios
In the question, it is given that the plane passes through (1,2,1)
So, the equation of the plane will be in the form,
A(x-1)+B(y-2)+C(z-1)=0
It is also given that the plane is perpendicular to give 2 planes.
So, their normal to the plane would be perpendicular to the normal of both planes.
So, the required normal is a cross-product of the normals of planes
x-y-z-10=0 and x-2y+z-2=0
i.e,
-3i-2j-k=0
so, the direction ratios,
A=-3, B=-2, C=-1
putting the direction ratios in the previous equation of the plane,
-3(x-1)-2(y-2)-1(z-1)=0
-3x+3-2y+4-z+1=0
-3x-2y-z+8=0
is the required equation of the plane
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