First slide

Analysis of two circles in circles

Question

Two circles $x^{2} + y^{2} = 6$ and $x^{2} + y^{2} - 6 x + 8 = 0$ are given. Then the equation of the circle through their point of intersection and the point $(1, 1)$ is

Moderate

A

$x^{2} + y^{2} - 6 x + 4 = 0$
B

$x^{2} + y^{2} - 3 x + 1 = 0$
C

$x^{2} + y^{2} - 4 x + 2 = 0$
D

none of these

Solution

Equation of the circle passing through the point of intersection of given circles $x^{2} + y^{2} - 6 x + 8 + λ (x^{2} + y^{2} - 6) = 0$ ….. $(1)$
This circle passes through $(1, 1)$ , then

$1 + 1 - 6 + 8 + λ (1 + 1 - 6) = 0 \Rightarrow λ = 1$

On putting the value of $λ$ in equation $(1)$

then $2 x^{2} + 2 y^{2} - 6 x + 2 = 0$

$\Rightarrow$ $x^{2} + y^{2} - 3 x + 1 = 0$ .

This is the required equation of circle.

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