First slide

Definition of a circle

Question

The intercept on the line y = x by the circle $x^{2} + y^{2} - 2 x = 0$ is AB. The equation of the circle with AB as diameter, is

Moderate

A

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

$x^{2} + y^{2} = x + y$
C

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

none of these

Solution

The equation of the circle passing through the intersect-ion of the circle $x^{2} + y^{2} - 2 x = 0$ and the line ( $A B$ whose equation is $y = x$ ), is
$\begin{aligned} x^{2} + y^{2} - 2 x + λ (y - x) = 0 \\ \Rightarrow x^{2} + y^{2} - x (2 + λ) + λ y = 0 \end{aligned}$
Line y = x will be a diameter of this circle, if it passes through
the centre $(\frac{2 + λ}{2}, - \frac{λ}{2})$
$∴ - \frac{λ}{2} = \frac{2 + λ}{2} \Rightarrow λ = - 1 .$
Putting $λ = - 1 in (i)$ we get
$x^{2} + y^{2} - x - y = 0$ as the equation of the required circle

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