The point (2, 3) is a limiting point of a coa | vedclass

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Question

(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

Vedclass · Accepted Answer

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

Vedclass · Answer

(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}} ight){m{ }},\;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 $

$\Rightarrow \lambda = \frac{{18}}{{13}},\;2$

$	herefore $The limiting points are (2, 3) and $\left[ {\frac{{18}}{{13}},\;\frac{3}{2}\left( {\frac{{18}}{{13}}} ight)} ight]$

or $\left( {\frac{{18}}{{13}},\;\frac{{27}}{{13}}} ight)$.

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

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