Line xcos α + ysin α = p will be a tangent to conic frac{{{x^2}}}{{{a^2}}} + frac{{{y^2}}}{{{b^2}}

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10-2. Parabola, Ellipse, Hyperbola

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The line $x\cos \alpha + y\sin \alpha = p$ will be a tangent to the conic $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$, if

A

${p^2} = {a^2}{\sin ^2}\alpha + {b^2}{\cos ^2}\alpha $

B

${p^2} = {a^2} + {b^2}$

C

${p^2} = {b^2}{\sin ^2}\alpha + {a^2}{\cos ^2}\alpha $

D

None of these

Solution

(c) $y = – x\cot \alpha + \frac{p}{{\sin \alpha }}$ is

tangent to $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1,$

if $\frac{p}{{\sin \alpha }} = \pm \sqrt {{b^2} + {a^2}{{\cot }^2}\alpha } $

or ${p^2} = {b^2}{\sin ^2}\alpha + {a^2}{\cos ^2}\alpha $.

Standard 11

Mathematics

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