First slide

Analysis of two circles in circles

Question

The equation of the smallest circle passing through the points of intersection of the line $x + y - 1 = 0$ and the circle $x^{2} + y^{2} = 9$

Easy

A

$x^{2} + y^{2} + x + y - 8 = 0$
B

$x^{2} + y^{2} - x - y - 8 = 0$
C

$x^{2} + y^{2} - x + y - 8 = 0$
D

$x^{2} + y^{2} - x - y + 8 = 0$

Solution

Given circle $x^{2} + y^{2} = 9....... (1)$ and line if $x + y - 1 = 0............ (2)$
The point of intersection of $(1) & (2)$ is $(1) + λ (2) = 0$

$\Rightarrow x^{2} + y^{2} - 9 + λ (x + y - 1) = 0$

$\Rightarrow x^{2} + y^{2} + λ x + λ y - λ - 9 = 0......... (3)$

Centre $(\frac{- λ}{2}, \frac{- λ}{2})$

Since equation $(3)$ is a smallest circle then equation $(2)$ is a diameter of the circle

$\Rightarrow C e n t r e (\frac{- λ}{2}, \frac{- λ}{2})$ lies on $(2)$

$\frac{- λ}{2}, \frac{- λ}{2} - 1 = 0 \Rightarrow λ = - 1$

Substituting $λ = - 1$ in equation $(3)$

$\Rightarrow x^{2} + y^{2} - 9 - (x + y - 1) = 0$

$\Rightarrow x^{2} + y^{2} - x - y - 8 = 0$

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