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If x and y are connected parametrically by the equations given in Exercise x=a \sec \theta, y=b \tan \theta without eliminating the parameter, find dy/dx.
CBSE | Class 12 | Continuity and Differentiability | Exercise 5.6 | Maths | NCERT
If x and y are connected parametrically by the equations given in Exercise x=a \sec \theta, y=b \tan \theta without eliminating the parameter, find dy/dx.

Solution:

The provided functions are x=a \sec \theta and y=b \tan \theta

\frac{d x}{d \theta}=a \sec \theta \tan \theta and

\frac{d y}{d \theta}=\sec ^{2} \theta

So now,

\frac{d y}{d x}=\frac{d y / d \theta}{d x / d \theta}=\frac{b \sec ^{2} \theta}{a \sec \theta \tan \theta}

=\frac{b \sec \theta}{a \tan \theta}

\begin{aligned} & \frac{b \cdot \frac{1}{\cos \theta}}{a \cdot \frac{\sin \theta}{\cos \theta}} \\=& \frac{b}{\cos \theta} \cdot \frac{\cos \theta}{\sin \theta} \\=& \frac{b}{a \sin \theta} \\=& \frac{b}{a} \operatorname{cosec} \theta \end{aligned}

i

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