Question

If st =1, then the tangent at P and the normal at S to the parabola meet at a point whose ordinate is
(A) $\dfrac{{\left( {{{\rm{t}}^2} + 1} \right)}}{{2{{\rm{t}}^3}}}$
(B) $\dfrac{{{\rm{a}}{{\left( {{{\rm{t}}^2} + 1} \right)}^2}}}{{2{{\rm{t}}^3}}}$
(C) $\dfrac{{{\rm{a}}{{\left( {{{\rm{t}}^2} + 1} \right)}^2}}}{{{{\rm{t}}^3}}}$
(D) $\dfrac{{{\rm{a}}{{\left( {{{\rm{t}}^2} + 2} \right)}^3}}}{{{{\rm{t}}^3}}}$

Answer 1

We know that equation of a tangent of a parabola at ${\rm{p}}:{\rm{y}} = {\rm{x}} + {\rm{a}}{{\rm{t}}^2}$. Also, equation of normal of parabola at point ${\rm{s}}:{\rm{y}} + \dfrac{{\rm{x}}}{{\rm{t}}} = \dfrac{{2{\rm{a}}}}{{\rm{t}}} + \dfrac{{\rm{a}}}{{{{\rm{t}}^3}}}$. Here, we have 2 equations and one unknown which is y. so we can easily get it’s value.

Complete step by step solution:
Given
st = 1
According to the question.
We know that
Equation of tangent at P to parabola;
${\rm{ty}} = {\rm{x}} + {\rm{a}}{{\rm{t}}^2}$
$\Rightarrow {\rm{y}} = \dfrac{{\rm{x}}}{{\rm{t}}} + \dfrac{{{\rm{a}}{{\rm{t}}^2}}}{{\rm{t}}}$
$\Rightarrow {\rm{y}} = \dfrac{{\rm{x}}}{{\rm{t}}} + {\rm{at}}$ (2)
And,
Also, we know that normal equation at S to the parabola.
then, equation of normal, ${\rm{y}} + \dfrac{{\rm{x}}}{{\rm{t}}} = \dfrac{{2{\rm{a}}}}{{\rm{t}}} + \dfrac{{\rm{a}}}{{{{\rm{t}}^3}}}$ (1)
From equation 1 and 2; we get
$2{\rm{y}} = {\rm{at}} + \dfrac{{2{\rm{a}}}}{{\rm{t}}} + \dfrac{{\rm{a}}}{{{{\rm{t}}^3}}}$
$\Rightarrow {\rm{y}} = \dfrac{{{\rm{at}}}}{2} + \dfrac{{2{\rm{a}}}}{{2{\rm{t}}}} + \dfrac{{\rm{a}}}{{2{{\rm{t}}^3}}}$
$\Rightarrow {\rm{y}} = \dfrac{{{\rm{at}}}}{2} + \dfrac{{\rm{a}}}{{\rm{t}}} + \dfrac{{\rm{a}}}{{2{{\rm{t}}^3}}}$
on, adding, we get
${\rm{y}} = \dfrac{{{\rm{a}}{{\rm{t}}^4} + 2{\rm{a}}{{\rm{t}}^2} + {\rm{a}}}}{{2{{\rm{t}}^3}}}$

Hence, the ordinate is $\dfrac{{{\rm{a}}\left( {{{\rm{t}}^2} + 1} \right)}}{{2{{\rm{t}}^3}}}$

Note:
The line perpendicular to the tangent of the parabola at the point of contact is called the normal. And the line touches the parabola at one point, then the line is called tangent. Another way to solve this problem is to find tangent and normals from our classical method which is by differentiating.