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____ [38]
3 years ago
6

If (8, −9) and (6, −3) are the endpoints of the diameter of a circle, what is the equation of the circle?

Mathematics
1 answer:
Umnica [9.8K]3 years ago
7 0

(x-7)^{2}  + (y+6)^{2}  = 10 is the equation of the required circle.

Step-by-step explanation:

Step 1 :

Let A = (8,-9) and B = (6,-3) be the endpoints of the diameter of the given circle.

We need to determine the circle's equation with this diameter

Step 2 :

The mid point of the diameter will be center of the required circle.

Let M = (x,y) be the mid point of the circle.

So  we have x = (8+6)÷ 2 , y = ((-9) + (-3)) ÷ 2

M = ( 7, -6) is the midpoint of the diameter and the center of the circle.

Step 3 :

The radius of the circle is  distance between  midpoint and any one of the end point of the diameter that is  distance between  point A(8,-9) and  M ( 7,-6)

Distance between the 2 points (x_{1} ,y_{1})  and  (x_{2} ,y_{2}) is given by

\sqrt{(x_{2}-x_{1})^{2}+ (y_{2}-y_{1})^{2} }

So distance between  A(8,-9) and  M ( 7,-6)  =  \sqrt{(7-8)^{2}+ (-6-(-9))^{2} }

                                                                        =  \sqrt{(1)^{2}+ (3)^{2} } =  \sqrt{10 }

Hence the radius r is \sqrt{10}

Step 4 :

The equation of circle with center (a,b) = (7,-6) and radius r = \sqrt{10} is given by

(x-a)^{2}  + (y-b)^{2} = r^{2}

=> (x-7)^{2}  + (y-(-6))^{2}  = (\sqrt{10} )^{2}

=> (x-7)^{2}  + (y+6)^{2}  = 10 is equation of the required circle.

Step 5 :

Answer :

(x-7)^{2}  + (y+6)^{2}  = 10  is  equation of the required circle.

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