First slide

Definition of a circle

Question

The equation of the circle which is touched by $y = x$ , has its centre on the positive direction of the $x - a x i s$ and cuts off a chord of length 2 units along the line $\sqrt{3} y - x = 0,$ is

Moderate

A

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

$x^{2} + y^{2} - 4 x + 1 = 0$
C

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

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

Solution

Since the required circle has its centre on x-axis, So, let the coordinates of the centre be $(a, 0) .$ The circle touches $y = x .$ Therefore,
Radius = Length of the . from $(a, 0)$ on $x - y = 0$
$\Rightarrow Radius = \frac{a}{\sqrt{2}}$
The circle cuts off a chord of length 2 units along $x - \sqrt{3} y = 0$
$∴ {(\frac{a}{\sqrt{2}})}^{2} = 1^{2} + {(\frac{a - \sqrt{3} \times 0}{\sqrt{1^{2} + (\sqrt{3})^{2}}})}^{2} \Rightarrow \frac{a^{2}}{2} = 1 + \frac{a^{2}}{4} \Rightarrow a = 2$
Thus, centre of the circle is at $(2, 0)$ and radius $= \frac{a}{\sqrt{2}} = \sqrt{2}$
Hence, its equation is $x^{2} + y^{2} - 4 x + 2 = 0$

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