question_answer

A curve is represented by the equations \[x={{\sec }^{2}}\]t and\[y=\cot \,t\], where t is a parameter. If the tangent at the point P on the curve where \[t=\pi /4\] meets the curve again at the point Q, Then \[\left| PQ \right|\] is equal to

A) \[\frac{5\sqrt{3}}{2}\]   

B) \[\frac{5\sqrt{5}}{2}\]

C) \[\frac{2\sqrt{5}}{3}\]   

D) \[\frac{3\sqrt{5}}{2}\]

Correct Answer: D

Solution :

[d] Eliminating t gives \[{{y}^{2}}(x-1)=1.\] Equation of the tangent at \[P(2,1)\] is \[x+2y=4.\] Solving with curve \[x=5\]and\[y=-1/2\], we get \[Q\equiv (5,-1/2)\] or \[PQ=\frac{3\sqrt{5}}{2}\]