Conic Sections Question 507
Question: If a line $ y=3x+1 $ cuts the parabola $ x^{2}-4x-4y+20=0 $ at A and B, then the tangent of the angle subtended by line segment AB, at the origin is
Options:
A) $ 8\sqrt{3}/205 $
B) $ 8\sqrt{3}/209 $
C) $ 8\sqrt{3}/215 $
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
[b] The joint equation of OA and OB is $ x^{2}-4x(y-3x)-4y(y-3x)+20{{(y-3x)}^{2}}=0 $
Making the equation of the parabola homogeneous using a straight line.
We get $ x^{2}(1+12+180)-y^{2}(4-20)-xy(4-12+120)=0 $ or $ 193x^{2}+16y^{2}-112xy=0 $
$ \tan \theta =\frac{2\sqrt{h^{2}-ab}}{a+b} $
$ =\frac{2\sqrt{56^{2}-193\times 16}}{193+16}=\frac{8\sqrt{3}}{209} $
BETA
