question_answer

The length of the chord of the parabola \[{{x}^{2}}=4y\] passing through the vertex and having slope \[\cot \alpha \] is

A) \[4\cos \alpha \cos \text{e}{{\text{c}}^{2}}\alpha \]       

B) \[4\tan \alpha \sec \alpha \]

C) \[4\sin \alpha {{\sec }^{2}}\alpha \]                        

D)  None of these

Correct Answer: A

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\[\Delta APB\],             \[AP=\sqrt{4{{t}^{2}}+{{t}^{4}}}\]                 \[=t\sqrt{4+{{t}^{2}}}\] \[\therefore \]  \[AP=2\cot \alpha \sqrt{4(1+{{\cot }^{2}}\alpha )}\] \[=2\cot \alpha \sqrt{4\cos \text{e}{{\text{c}}^{2}}\alpha }=4\cot \alpha \cos \text{ec}\alpha \] \[=4\frac{\cos \alpha }{\sin \alpha }\cos \text{ec}\alpha =4\cos \alpha \cos \text{e}{{\text{c}}^{2}}\alpha \]