Question

The angle between the tangents drawn from the point $(1,4)$ to the parabola ${y^2} = 4x$ is
A.$0$
B.$\dfrac{\pi }{6}$
C.$\dfrac{\pi }{4}$
D.$\dfrac{\pi }{3}$

Answer 1

Hint : A parabola is a U-shaped plane curve where any point is at an equal distance from a fixed point (known as the focus) and from a fixed straight line which is known as the directrix. If the directrix is parallel to the y-axis in the standard equation of a parabola is given as: ${y^2} = 4ax$. If the parabola is sideways i.e., the directrix is parallel to x-axis, the standard equation of a parabola becomes, ${x^2} = 4ay$.

Complete step-by-step answer :
Tangent, in geometry, is a straight line (or smooth curve) that touches a given curve at one point; at that point the slope of the curve is equal to that of the tangent.
Any tangent to the parabola ${y^2} = 4ax$ is given by $y = mx + \dfrac{a}{m}$.
We know that any quadratic equation in the variable $x$ is of the form $a{x^2} + bx + c = 0$ .
Sum of the roots$= - \dfrac{b}{a}$.
Product of roots $= \dfrac{c}{a}$
Angle between two lines whose slopes are given is given by:
$\tan \theta = \left| {\dfrac{{{m_1} - {m_2}}}{{1 + {m_1}{m_2}}}} \right|$
Where ${m_1}$and ${m_2}$are the respective slopes.
We know that any tangent to the parabola ${y^2} = 4x$is $y = mx + \dfrac{1}{m}$.
As it passes through the point $(1,4)$ so it satisfies the above equation.
Therefore $4 = m + \dfrac{1}{m}$
Simplifying this equation we get ${m^2} - 4m - 1 = 0$
This is a quadratic equation in terms of$m$. Let ${m_1}$and ${m_2}$be the required roots.
Hence we have ${m_1} + {m_2} = 4$ and ${m_1}{m_2} = 1$
Now consider ${\left( {{m_1} - {m_2}} \right)^2} = {\left( {{m_1} + {m_2}} \right)^2} - 4{m_1}{m_2}$
Substituting the values we get ,
$\left| {{m_1} - {m_2}} \right| = 2\sqrt 3$
Therefore $\tan \theta = \left| {\dfrac{{{m_1} - {m_2}}}{{1 + {m_1}{m_2}}}} \right|$
$= \dfrac{{2\sqrt 3 }}{2} = \sqrt 3$
Therefore we get $\theta = {\tan ^{ - 1}}\sqrt 3 = \dfrac{\pi }{3}$
Therefore option ( $4$ ) is the correct answer.
So, the correct answer is “Option 4”.

Note : If the directrix is parallel to the y-axis in the standard equation of a parabola is given as: ${y^2} = 4ax$. If the parabola is sideways i.e., the directrix is parallel to x-axis, the standard equation of a parabola becomes, ${x^2} = 4ay$.