The equation of the circle is [tex](x - 1)^{2} + (y - 1)^{2} = \frac{52}{4}[/tex]
What is circle and its equation?
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]
What is distance formula?
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]
What is mid point formula?
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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