Conic-Sections Question 550
Question: If tangents are drawn from any point on the line $ x+4a=0 $ to the parabola $ y^{2}=4ax, $ then their chord of contact subtends angle at the vertex equal to
Options:
A) $ \frac{\pi }{4} $
B) $ \frac{\pi }{3} $
C) $ \frac{\pi }{2} $
D) $ \frac{\pi }{6} $
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Answer:
Correct Answer: C
Solution:
[c] Let $ R(-4a,k) $ be any point on the line $ x=-4a. $ The equation of chord of contact PQ w.r.t. $ P(-4a,k) $ is $ y.k=2a(x-4a) $ ?. (1) Making equation of parabola $ y^{2}=4ax $ Homogeneous using (1), we get $ y^{2}=4ax( \frac{2ax-yk}{8a^{2}} ) $
$ \Rightarrow 8a^{2}x^{2}-8a^{2}y^{2}-4akxy=0 $ This represents the pair of straight lines AP and AQ. Since coefficient of $ x^{2}+ $ coefficient of $ y^{2}=0\therefore \angle PAQ=90{}^\circ $ i.e., chord of contact PQ subtends a right angle at the vertex.
BETA
