Question

Example 4: Find the equation of the circle which passes through the points (2-2). and
(3,4) and whose centre lies on the line x + y = 2​

Answer 1

Answer:

Step-by-step explanation:

Let the equation of circle be $x^{2} +y^{2}+2gx+2fy+c=0$

So, the centre of the above circle is C(-g,-f) and radius is $r = \sqrt{{g^{2} + f^{2} - c }}$

So, the two given points lie on this circle.

i.e P(2,-2)  Q(3,4)

By substituting the values of the points P and Q on the circle we get the following equations

$4g-4f+8+c=0$ $-------(i)$

$6g+8f+25+c=0------(ii)$

Given the centre of the circle lies on the line $x + y = 2.$$-----(iii)$

i.e  C(-g,-f) lies on (iii)

So, by substituting C in (iii), we get

$g + f + 2 = 0----(iv)$

So, By subtracting (i) from (ii) we get,

$2g + 12f + 17 = 0----(v)$

By solving (iv) and (v) we get

$g = \frac{-7}{10} and f = \frac{-13}{10}$

So by substituting the values in (i) we get

$c = \frac{-52}{5}$

Finally, the equation of circle is

$5x^{2} +5y^{2}-7x-13y-52=0$.