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, \[{x^2} = 4ay\].