- Home
- Standard 11
- Mathematics
The point $(2, 3)$ is a limiting point of a coaxial system of circles of which ${x^2} + {y^2} = 9$ is a member. The co-ordinates of the other limiting point is given by
$\left( {\frac{{18}}{{13}},\frac{{27}}{{13}}} \right)$
$\left( {\frac{9}{{13}},\frac{6}{{13}}} \right)$
$\left( {\frac{{18}}{{13}}, - \frac{{27}}{{13}}} \right)$
$\left( { - \frac{{18}}{{13}}, - \frac{9}{{13}}} \right)$
Solution
(a) ${(x – 2)^2} + {(y – 3)^2} = 0$
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){\rm{ }},\;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 $\left[ {\frac{{18}}{{13}},\;\frac{3}{2}\left( {\frac{{18}}{{13}}} \right)} \right]$
or $\left( {\frac{{18}}{{13}},\;\frac{{27}}{{13}}} \right)$.
