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Home  >>  CBSE XI  >>  Math  >>  Straight Lines
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State whether the following statement is true or false and justify : Equation of the line passing through the point $(a \cos^3 \theta, a \sin^3 \theta)$ and perpendicular to the line $x \sec \theta + y \: cosec \theta = a $ is $ x \cos \theta - y \sin \theta = a \sin 2 \theta $

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Toolbox:
  • Equation of a line passing through a point $(x_1, y_1)$ and having slope $m$ is $y-y_1=m(x-x_1)$
  • If two lines are perpendicular then the product of their slope is -1.
Step 1 :
Equation of the given line is
$x \sec \theta + y \: cosec \theta = a$
Slope of this line is
$ -\large\frac{\sec \theta}{cosec \theta } $$= -\large\frac{\sin \theta }{\cos \theta }$
Hence slope of the line perpendicular to the above line is $ \large\frac{\cos \theta}{\sin \theta}$
It passes through the point $( a \cos^3 \theta , a \sin^3 \theta )$
$ \therefore $ Equation of the required line is
$(y- a \sin^3 \theta ) = \large\frac{\cos \theta }{\sin \theta}$$ (x - a \cos^3 \theta )$
$ \Rightarrow \sin \theta (y- a \sin^3 \theta ) = \cos \theta (x - a \cos^3 \theta )$
$ \Rightarrow y \sin \theta - a \sin^4 \theta = x \cos \theta - a \cos^4 \theta $
$ \Rightarrow x \cos \theta - y \sin \theta = a \cos^4 \theta - a \sin^4 \theta $
$ \Rightarrow x \cos \theta - y \sin \theta = a ( cos^2 \theta + \sin^2 \theta ) ( \cos^2 \theta - \sin^2 \theta )$
But $ \cos^2 \theta + \sin^2 \theta = 1 $ and $ \cos^2 \theta - \sin^2 \theta = \cos 2\theta $
$ \therefore x \cos \theta - y \sin \theta = a \cos 2 \theta $
Hence the statement is true.
answered Jul 8, 2014 by thanvigandhi_1
 

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