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?

• A. ${30}^o$
• B. ${60}^o$
• C. ${90}^o$
• D. ${120}^o$

Answer 1

The correct option is C ${90}^o$

Let us assume that the ends of double ordinate are $P(a{t}^2,2at)$ and $P^{\prime }=(a{t}^2,-2at)$

Now, we also have $P{P}^{\prime }=8a$ $sqrt(a{t}^2-a{t}^2{)}^2+(2at+2at{)}^2=8a$

$\therefore t=2$

Hence $P=(a{t}^2,2at)=(4a,4a)$ and $P^{\prime }=(a{t}^2,-2at)=(4a,-4a)$

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 $O{P}^{\prime }$ and $OP$ = ${\tan }^{-1}\frac{m_1-m_2}{1+m_1{m}_2}={\tan }^{-1}\frac{2}{0}=\frac{\pi }{2}$

Hence, the line joining the origin to ends of the double ordinate will be perpendicular to each other.