Conic Sections Question 401
Question: If a double ordinate of the parabola $ y^{2}=4ax $ be of length $ 8a $ , then the angle between the lines joining the vertex of the parabola to the ends of this double ordinate is
Options:
A) 30o
B) 60o
C) 90o
D) 120o
Show Answer
Answer:
Correct Answer: C
Solution:
The correct option is C 90oLet us assume that the ends of double ordinate are $P(at^2,2at)$ and $P′=(at^2,−2at)$
Now, we also have PP′=$8a sqrt(at^2−at^2)^2+(2at+2at)^2$=8a
∴t=2
Hence P=$(at^2,2at)$=(4a,4a) and P′=$(at^2,−2at)$=(4a,−4a)
Now, the vertex is O(0,0) for the parabola.
slope of OP=$m_1=\frac{4a−0}{4a−0}=1$
slope of OP’=$m_2=\frac{4a−0}{−4a−0}=−1$
Angle between the lines OP′ and OP = $tan^{−1}\frac{m_1−m_2}{1+m_1m_2}=tan^{−1}\frac{2}{0}=\frac{π}{2}$
Hence, the line joining the origin to ends of the double ordinate will be perpendicular to each other.
BETA
