question_answer

The tangent drawn at any point P to the parabola \[{{y}^{2}}=4ax\] meets the directrix at the point K, then the angle which KP subtends at its focus is                                                                                      [RPET 1996, 2002]

A)            30o 

B)            45o

C)            60o 

D)            90o

Correct Answer: D

Solution :

           Here, \[P(a{{t}^{2}},\,2at)\] and S(a, 0).            If the tangent at P, \[ty=x+a{{t}^{2}},\] meets the directrix            \[x=-a\,\,\text{at}\,\,k,\] then \[k=\left( -a,\,\frac{a{{t}^{2}}-a}{t} \right)\]            \[{{m}_{1}}=\] slope of \[SP=\frac{2at}{a({{t}^{2}}-1)}\]            \[{{m}_{2}}=\] slope of \[SK=\frac{a({{t}^{2}}-1)}{-2at}\]            Clearly \[{{m}_{1}}{{m}_{2}}=-1\],  \[\therefore \,\angle \,PSK={{90}^{o}}.\]