JEE Main & Advanced
Mathematics
Conic Sections
Question Bank
Parabola
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 \[{{m}_{1}}{{m}_{2}}=-1\], \[\therefore \,\angle \,PSK={{90}^{o}}.\]