Find the radical center of the circles $x^{2} + y^{2} + 4 x - 7 = 0, 2 x^{2} + 2 y^{2} + 3 x + 5 y - 9 = 0, x^{2} + y^{2} + y = 0$ .

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Solution

Given,
$x^{2} + y^{2} + 4 x - 7 = 0$ ... $(1)$
$2 x^{2} + 2 y^{2} + 3 x + 5 y - 9 = 0$ ... $(2)$
$x^{2} + y^{2} + y = 0$ ... $(3)$
eqn $(2) - 2 \times$ eqn $(1)$ gives,
$- x + y = 1$ ... $(A)$
eqn $(2) - 2 \times$ eqn $(3)$ gives,
$x + y = 3$ ... $(B)$
solving $(A)$ and $(B)$ , we get,
$y = 2$
$- x + y = 1 \Rightarrow - x + 2 = 1 \Rightarrow x = 1$
$∴ (1, 2)$ is the radical center of the given circles.

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Similar questions

If (h, k) is a point from which the tangents to the three circles $x^{2} + y^{2} - 4 x + 7 = 0$ , $2 x^{2} + 2 y^{2} - 3 x + 5 y + 9 = 0$ and $x^{2} + y^{2} + y = 0$ are equal in length. Find the value of h + k ___ .

Q. Find the equation of the circle which cuts the following circles orthogonally.
(i)

x^{2} + y^{2} + 4 x - 7 = 0

2 x^{2} + 2 y^{2} + 3 x + 5 y - 9 = 0

x^{2} + y^{2} + y = 0

Q.
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x^{2} + y^{2} = 9, x^{2} + y^{2} - 2 x - 2 y - 5 = 0

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and

2 x^{2} + 2 y^{2} + 3 x + 8 y + 2 c = 0

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Q. The radical axis of circles

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and

x^{2} + y^{2} - 5 x + 5 y - 6 = 0

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Find the radical center of the circles x2+y2+4x−7=0,2x2+2y2+3x+5y−9=0,x2+y2+y=0.

Given,x2+y2+4x−7=0 ...(1)2x2+2y2+3x+5y−9=0 ...(2)x2+y2+y=0 ...(3)eqn(2)−2×eqn(1) gives,−x+y=1 ...(A)eqn(2)−2×eqn(3) gives,x+y=3 ...(B)solving (A) and (B), we get,y=2−x+y=1⇒−x+2=1⇒x=1∴(1,2) is the radical center of the given circles.

Find the radical center of the circles $x^{2} + y^{2} + 4 x - 7 = 0, 2 x^{2} + 2 y^{2} + 3 x + 5 y - 9 = 0, x^{2} + y^{2} + y = 0$ .