Question

Show that equation of the line passing through (acos^3 theeta,asin^3theeta) and perpendicular to the line x sec theeta + ycosec theeta = a is x cos theeta -y sin theeta = a cos2 theeta

Answer 1

$xcos\theta - y sin \theta = a cos2\theta$

Step-by-step explanation:

Given equation of line is

$xsec\theta +ycosec\theta = a$

⇒$y=\frac{a}{cosec\theta }- \frac{xsec\theta }{cosec\theta }$

Therefore the slope of the line is$(m_{1} )$ $=- \frac{sec\theta }{cosec\theta }$

The slope of the line which is perpendicular to the given line is =$\frac{-1}{m_{1} }$

                                                                                                      $= \frac{cosec\theta }{sec\theta }$

                                                                                                      $= \frac{cos\theta }{sin\theta }$

Therefore the equation of the straight line which passes through the point $(acos^3\theta , asin^3\theta)$ and  slope is = $\frac{cos\theta }{sin\theta }$ is

$(y-asin^3\theta)= \frac{cos\theta }{sin\theta } (x-acos^3\theta)$

⇔$y sin \theta -asin\theta sin^3 \theta = x cos\theta - a cos\theta cos^3 \theta$

⇔$xcos\theta - y sin \theta = a cos^4\theta -asin^4\theta$

⇔$xcos\theta - y sin \theta = a(cos^2\theta +sin^2\theta) (cos^2\theta -sin^2\theta)$

⇔$xcos\theta - y sin \theta = a cos2\theta$(proved)

Answer 2

AnswEr:

The equation of a line perpendicular to the line x sec ∅ + y cos ∅ = a is

• $\sf \: x \: cosec \: \theta - y \: sec \: \theta + \lambda = 0 \: ( \lambda \: is \: a \: constant) \\ - - -(1.)$

The line passing through (a cos³ ∅, a sin³ ∅).

$\therefore \tt \: a \: cos {}^{3} \: \theta \: cosec \: \theta - a \: sin {}^{3} \theta \: sec \: \theta + \lambda = 0 \\ \\ \implies \tt \lambda = a( {sin}^{3} \: \theta \: sec \: \theta - {cos}^{3} \: \theta \: cosec \: \theta$

_______________________

Putting the value of $\lambda$ in (1), we get

$\implies \sf \frac{x}{sin \: \theta} - \frac{y}{cos \: \theta} + a( \frac{ {sin}^{3} \: \theta}{cos \: \theta} - \frac{ {cos}^{3} \: \theta }{sin \:\theta } ) = 0 \\ \\ \sf \implies \: x \: cos \: \theta - y \: sin \: \theta + a( {sin}^{4} \: \theta - {cos}^{4} \: \theta) = 0 \\ \\ \implies \sf \: x \: cos \: \theta - y \: sin \: \theta + a( {sin}^{2} \: \theta + {cos}^{2} \: \theta) \\ \sf( {sin}^{2} \: \theta - {cos}^{2} \: \theta) = 0 \\ \\ \implies \sf \: x \: cos \: \theta - y \: sin \: \theta - a( {cos}^{2} \: \theta - {sin}^{2} \: \theta) = 0 \\ \\ \sf \implies \: x \: cos \: \theta - y \: sin \: \theta - a \: cos \: 2 \: \theta = 0 \\ \\ \sf \implies \: x \: cos \: \theta - y \: sin \: \theta = a \: cos \: 2 \: \theta \:$