First slide

Definition of a circle

Question

PQ is a chord of the circle $x^{2} + y^{2} - 2 x - 8 = 0$ whose midpoint is (2, 2). The circle passing through P, Q and (1, 2) is

Moderate

A

$x^{2} + y^{2} - 7 x + 10 y + 28 = 0$
B

$x^{2} + y^{2} - 7 x - 10 y + 22 = 0$
C

$x^{2} + y + 7 x - 10 y + 22 = 0$
D

$x^{2} + y^{2} + 7 x + 10 y - 22 = 0$

Solution

The equation of the chord of the circle $x^{2} + y^{2} - 2 x - 8 = 0$ having (2, 2) as its mid-point is
$\begin{array}{l} 2 x + 2 y - (x + 2) - 8 = 4 + 4 - 4 - 8 [Using: S^{'} = T] \\ \Rightarrow x + 2 y - 6 = 0 \end{array}$
The equation of a circle passing through P and Q is
$x^{2} + y^{2} - 2 x - 8 + λ (x + 2 y - 6) = 0$ ... (i)
It passes through (1, 2).
$∴ 1 + 4 - 2 - 8 + λ (1 + 4 - 6) = 0 \Rightarrow λ = - 5$
Putting the value of $λ$ in (i), we obtain
$x^{2} + y^{2} - 7 x - 10 y + 22 = 0$
as the equation of the required circle.

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