Question

The length of the chord of the parabola $ {{x}^{2}}=4ay $ passing through the vertex and having slope $ \tan \alpha $ is

• A. $ 4a\cos \text{ec}\alpha \cdot \cot \alpha $
• B. $ 4a\tan \alpha \cdot \sec \alpha $
• C. $ 4a\cos \alpha \cdot \cot \alpha $
• D. $ 4a\sin \alpha \cdot \tan \alpha $

Answer 1

Answer: (B)

Let $ A $ be the vertex and $ AP $ be a chord of $ {{x}^{2}}=4ay $ such that slope of $ AP $ is $ \tan \alpha $ .
Let the coordinates of $ P $ be $ (2at,\,\,a{{t}^{2}}) $ then, Slope of $ AP=\frac{a{{t}^{2}}}{2at}=\frac{t}{2} $
$ \Rightarrow $ $ \tan \alpha =\frac{t}{2} $
$ \Rightarrow $ $ t=2\tan \alpha $ Now, $ AP=\sqrt{{{(2at-0)}^{2}}+{{(a{{t}^{2}}-0)}^{2}}} $
$ =at\sqrt{4+{{t}^{2}}}=2a\tan \alpha \sqrt{4+4{{\tan }^{2}}\alpha } $
$ =4a\tan \alpha \cdot \sec \alpha $