Question

The angle between the tangents at the ends of a focal chord of a parabola is
A.$90^\circ$
B.$45^\circ$
C.$60^\circ$
D.N.O.T.

Answer 1

Hint: We will use the standard equation of a parabola to find the angle between the tangents at the ends of a focal chord of a parabola. We have to find the product of the slopes of the tangents in order to find the angle.

Complete step-by-step answer:
A chord of a parabola which passes through its focus is called a focal chord of the parabola. A parabola is formed when the constant ratio eccentricity is equals to 1.
Let us assume$\;{y^2} = 4ax$ to be the standard equation of a parabola.
We will let $P(at_1^2,2a{t_1})$ and $Q(at_2^2,2a{t_2})$ be two points on the curve.
The equation of PQ will therefore be $(y - 2a{t_1}) = \dfrac{2}{{{t_1} + {t_2}}}\left( {x - at_1^2} \right)$
As PQ is a focal chord, it passes through $S(a,0)$,

$$\Rightarrow - 2a{t_1} = \dfrac{2}{{{t_1} + {t_2}}}(a - at_1^2) \\\ \Rightarrow - 2a{t_1}({t_1} + {t_2}) = 2(a - at_1^2) \\\ \Rightarrow - 2at_1^2 - 2a{t_1}{t_2} = 2a - 2at_1^2 \\\ \Rightarrow - 2a{t_1}{t_2} = 2a \\\ \Rightarrow {t_1}{t_2} = - 1 \\\$$

We will now let ${m_1}\& {m_2}$be the slopes of the tangents P and Q respectively.
${m_1}{m_2} = {t_1}{t_2} = - 1$
Therefore, the angle between the tangents at the ends of the focal chord of the parabola is $90^\circ$ because if the product of two slopes is $- 1$, then the two lines are said to be perpendicular.
Thus, the answer is option A.

Note: This question is an observation from a close examination of the four standard equations of the parabola and the results derived in relation to them. We have assumed the equation of the parabola to be $\;{y^2} = 4ax$ where the origin is the vertex and the x-axis is the axis of the parabola.