Q. The length of the chord of the parabola $ {{x}^{2}}=4y $ passing through the vertex and having slope $ \cot \alpha $ is
Jharkhand CECEJharkhand CECE 2008
Solution:
Let $ A $ be the vertex of the parabola and $ AP $ is chord of parabola such that slope of $ AP $ is $ \cot \alpha $ .
Let coordinates of $ P $ be $ (2t,\,\,{{t}^{2}}) $
which is a point on the parabola.
$ \therefore $ Slope of $ AP=\frac{t}{2} $
$ \Rightarrow $ $ \cot \alpha =\frac{t}{2} $
$ \Rightarrow $ $ t=2\cot \alpha $
In $ \Delta APB $ , $ AP=\sqrt{4{{t}^{2}}+{{t}^{4}}} $
$ =t\sqrt{4+{{t}^{2}}} $
$ \therefore $ $ AP=2\cot \alpha \sqrt{4(1+{{\cot }^{2}}\alpha )} $
$ =2\cot \alpha \sqrt{4\cos \text{e}{{\text{c}}^{2}}\alpha }=4\cot \alpha \cos \text{ec}\alpha $
$ =4\frac{\cos \alpha }{\sin \alpha }\cos \text{ec}\alpha =4\cos \alpha \cos \text{e}{{\text{c}}^{2}}\alpha $
