Question

The point (2, 3) is a limiting point of a co-axial system of circles of which $x^{2} + y^{2} = 9$is a member. The coordinates of the other limiting point is given by

• A. $\left( \frac{18}{13},\frac{27}{13} \right)$
• B. $\left( \frac{9}{13},\frac{6}{13} \right)$
• C. $\left( \frac{18}{13}, - \frac{27}{13} \right)$
• D. $\left( - \frac{18}{13}, - \frac{9}{13} \right)$

Answer 1

Answer: (A)

$\left( \frac{18}{13},\frac{27}{13} \right)$

Answer 2

Equation of circle with (2, 3) as limiting point is

$(x - 2)^{2} + (y - 3)^{2} = 0$ or $(x^{2} + y^{2} - 9) - 4x - 6y + 22 = 0$

or $(x^{2} + y^{2} - 9) - \lambda(2x + 3y - 11) = 0$ represents the family of co-axial circles.

$c = \left( \lambda,\frac{3\lambda}{2} \right),r = \sqrt{\lambda^{2} + \frac{9\lambda^{2}}{4} - 11\lambda + 9}$.

For limiting points r = 0

$\Rightarrow 13\lambda^{2} - 44\lambda + 36 = 0 \Rightarrow \lambda = \frac{18}{13},2$

$\therefore$ The limiting points are (2, 3) and $\left\lbrack \frac{18}{13},\frac{3}{2}\left( \frac{18}{13} \right) \right\rbrack$ or

$\left( \frac{18}{13},\frac{27}{13} \right)$