If P = (-2,-1) and Q = (4,3) are the
endpoints of the diameter of a circle,
find the equation of the circle.
(x - [?])2 + (y - [ ])2 = []

Answer:

Step-by-step explanation:

P(-2,-1) and Q(4,3)

average of x-coordinates = (-2+4)/2 = 1

average of y-coordinates = (-1+3)/2 = 1

midpoint of PQ: (1,1)

distance between midpoint and Q = √((4-1)²+(3-1)²) = √13

(x-1)² + (y-1)² = 13

The equation of the circle is [tex](x - 1)^{2} + (y - 1)^{2} = \frac{52}{4}[/tex]

A circle is a shape consisting of all points in a plane that are at a given distance from a given point (the Centre) Equivalently .

Equation of circle

The equation of circle provides an algebraic way to describe a circle, given the center and the length of the radius of a circle.

The equation of circle represents all the points that lie on the circumference of the circle .

The standard equation of a circle with center at [tex](h , k )[/tex] and radius r is

  [tex](x - h)^{2} + (y - k)^{2} = r^{2}[/tex]

The distance formula in coordinate geometry is used to calculate the distance between two given points.

The formula says the distance between two points ([tex]x_{1} ,y_{1}[/tex]), and ([tex]x_{2} , y_{2}[/tex])

[tex]Distance(D) = \sqrt{(x_{2} - x_{1} )^{2} + (y_{2} - y_{1} )^{2} }[/tex]

The Midpoint Formula:

The midpoint of two ends coordinates points,  ([tex]x_{1} ,y_{1}[/tex]), and ([tex]x_{2} , y_{2}[/tex]) is the point M can be found by using:

[tex]M = \frac{x_{1} + x_{2}}{2}, \ \ \frac{y_{1} + y_{2}}{2}[/tex]

According to the question

P = (-2,-1) and Q = (4,3) are the

endpoints of the diameter of a circle

Therefore, distance between point P and Q which is diameter of circle is

 [tex]Distance(D) = \sqrt{(x_{2} - x_{1} )^{2} + (y_{2} - y_{1} )^{2} }[/tex]

P(-2,-1) =  ([tex]x_{1} ,y_{1}[/tex])

Q (4,3) =  ([tex]x_{2} , y_{2}[/tex])

Now,

Diameter of circle = [tex]\sqrt{(4 - (-2) )^{2} + (3 - (-1) )^{2} }[/tex]

                             = [tex]\sqrt{(6 )^{2} + (4 )^{2} }[/tex]

                             = [tex]\sqrt{(36 + 16) }[/tex]

                             = [tex]\sqrt{52}[/tex]

As , radius  =  [tex]\frac{diameter }{2}[/tex]

       radius (r) =  [tex]\frac{\sqrt{52}}{2}[/tex]          

As we know The center of the circle separates the diameter into two equal segments called radii and radii are equal in a circle .

i.e mid point of coordinates of diameter are coordinates of circle .

[tex]M = \frac{x_{1} + x_{2}}{2}, \ \ \frac{y_{1} + y_{2}}{2}[/tex]

[tex]Centre of circle (h,k) = \frac{x_{1} + x_{2}}{2}, \ \ \frac{y_{1} + y_{2}}{2}[/tex]

[tex]h = \frac{x_{1} + x_{2}}{2}, \ \ k = \frac{y_{1} + y_{2}}{2}[/tex]

[tex]h = \frac{ -2 + 4}{2}, \ \ k = \frac{-1 + 3}{2}[/tex]

[tex]h = 1, \ \ k = 1[/tex]

Now , substitute the value in the equation of circle

[tex](x - h)^{2} + (y - k)^{2} = r^{2}[/tex]

[tex](x - 1)^{2} + (y - 1)^{2} = \frac{52}{4}[/tex]

Hence, the equation of the circle is [tex](x - 1)^{2} + (y - 1)^{2} = \frac{52}{4}[/tex]

To  know more about equation of circle here:

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