First slide

Definition of a circle

Question

The equation of the circle passing through the point $(- 1, 2)$ and having two diameters along the pair of lines
$x^{2} - y^{2} - 4 x + 2 y + 3 = 0$ , is

Moderate

A

$x^{2} + y^{2} - 4 x - 2 y + 5 = 0$
B

$x^{2} + y^{2} + 4 x + 2 y - 5 = 0$
C

$x^{2} + y^{2} - 4 x - 2 y - 5 = 0$
D

none of these

Solution

The combined equation of two diameters is
$\begin{array}{l} x^{2} - y^{2} - 4 x + 2 y + 3 = 0 \\ \Rightarrow (x - 2)^{2} - (y - 1)^{2} = 0 \Rightarrow (x + y - 3) (x - y - 1) = 0 \end{array}$
Thus, the two diameters are $x + y - 3 = 0$ and $x - y - l = C$
These two intersect at $(2, 1) .$
Thus, the required circle has its centre at $(2, 1)$ and passes through $(- 1, 2)$ . So, its equation is
$\begin{array}{l} (x - 2)^{2} + (y - 1)^{2} = (2 + 1)^{2} + (1 - 2)^{2} \\ \Rightarrow x^{2} + y^{2} - 4 x - 2 y - 5 = 0 \end{array}$

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