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.