First slide

Definition of a circle

Question

Circles are drawn through the point $(3, 0$ $)$ to cut an intercept of length $6$ units on the negative direction of the $x - a x i s$ . The equation of the locus of their centres, is

Moderate

A

$y = 0$
B

$y = x$
C

$x = 0$
D

$y = - x$

Solution

Let the equation of the circle be
$x^{2} + y^{2} + 2 g x + 2 f y + c = 0$
It passes through $(3, 0)$ . Therefore
$9 + 6 g + c = 0$
Circle (i) cuts an intercept of $6$ units on the negative direction of $x - a x i s .$
$∴ 2 \sqrt{g^{2} - c = 6 \Rightarrow g^{2} - c = 9}$
From (ii) and (iii), we get
$\begin{array}{l} 9 + 6 g + g^{2} = 9 \\ \Rightarrow g (g + 6) = 0 \Rightarrow g = 0 [∵ g > 0 ∴ g + 6 \neq 0] \end{array}$
Hence, the locus of $(- g, - f)$ is $- x = 0$ , i.e. $x = 0 .$

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