If frac{x}{α } + frac{y}{β } = 1 touches circle {x^2} + {y^2} = {a^2} , then point (1/α ,,1/β )

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10-1.Circle and System of Circles

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If $\frac{x}{\alpha } + \frac{y}{\beta } = 1$ touches the circle ${x^2} + {y^2} = {a^2}$, then point $(1/\alpha ,\,1/\beta )$ lies on a/an

A

Straight line

B

Circle

C

Parabola

D

Ellipse

Solution

(b) $y = – \frac{\beta }{\alpha }x + \beta $ touches the circle,

${\beta ^2} = {a^2}\left( {1 + \frac{{{\beta ^2}}}{{{\alpha ^2}}}} \right)$

==> $\frac{1}{{{\alpha ^2}}} + \frac{1}{{{\beta ^2}}}$

$= \frac{1}{{{a^2}}}$

Locus of $\left( {\frac{1}{\alpha },\frac{1}{\beta }} \right)$ is ${x^2} + {y^2}$

$= {\left( {\frac{1}{a}} \right)^2}$.

Standard 11

Mathematics

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