Question

If two tangents drawn from the point P to the parabola ${{y}^{2}}=12x$ be such that the slope of the tangent is double than the other, then ‘P’ lies on the curve
(A) $2{{y}^{2}}=9x$
(B) ${{y}^{2}}=9x$
(C) $2{{y}^{2}}=27x$
(D) ${{y}^{2}}=15x$

Answer 1

We are given the equation of a parabola and a point P through which two tangents from the parabola passes through. We are also given that the slope of one tangent is double than that of the other. We will first find the general equation of the tangent to a parabola ${{y}^{2}}=4ax$ using the equation, $y=mx+\dfrac{a}{m}$. We will substitute the values in this equation and we will assume that it passes through $\left( h,k \right)$ which is our point P. Hence, we will have the required equation.

Complete step-by-step solution:
According to the given question, we are given the equation of a parabola and a point P through which two tangents from the parabola passes through. We are also given that the slope of one tangent is double than that of the other.
The equation of the parabola given to us is,
${{y}^{2}}=12x$----(1)
If we compare it to the general equation of the parabola which is, ${{y}^{2}}=4ax$, we have,
$a=3$
Now, the equation for a tangent to the parabola is given by the expression,
$y=mx+\dfrac{a}{m}$
Substituting the value of ‘a’, we get,
$\Rightarrow y=mx+\dfrac{3}{m}$ -----(2)
So, this is the general expression of the tangent to the given parabola.
Now, we are given that the tangents pass through the point P. let us assume that the coordinates of P be $\left( h,k \right)$, so we get the new expression from equation (2) as,
$\Rightarrow k=mh+\dfrac{3}{m}$
Simplifying the above expression, we get the quadratic equation which is,
${{m}^{2}}h-km+3=0$-----(3)
So, there are two slopes for the two tangents.
We are given that the slope of one of the tangents is twice the other, that is, we can write,
${{m}_{1}}=2{{m}_{2}}$----(4)
where ${{m}_{1}}$ and ${{m}_{2}}$ are two roots of the obtained quadratic equation.
Equation (3) is a quadratic equation and the roots can be found as,
Sum of the roots, ${{m}_{1}}+{{m}_{2}}=\dfrac{-(-k)}{h}=\dfrac{k}{h}$---(5)
Also, product of the roots, ${{m}_{1}}.{{m}_{2}}=\dfrac{3}{h}$----(6)
We will now substitute equation (4) in equation (6), we get,
$\Rightarrow 2{{m}_{2}}^{2}=\dfrac{3}{h}$-----(7)
Substituting equation (4) in equation (5), we get,
$\Rightarrow 3{{m}_{2}}=\dfrac{k}{h}$----(8)
Which ion rearranging, we get,
$\Rightarrow {{m}_{2}}=\dfrac{k}{3h}$----(9)
We will now be substituting the equation (9) in equation (7), we get,
$\Rightarrow 2{{\left( \dfrac{k}{3h} \right)}^{2}}=\dfrac{3}{h}$
Squaring the terms, we get the new expression as,
$\Rightarrow \dfrac{2{{k}^{2}}}{9{{h}^{2}}}=\dfrac{3}{h}$
Cancelling out the similar terms, we get,
$\Rightarrow 2{{k}^{2}}=27h$
Now, replacing the ‘h’ and ‘k’ with ‘x’ and ‘y’, we get,
$2{{y}^{2}}=27x$
Therefore, the correct option is (C) $2{{y}^{2}}=27x$.

Note: The parabola and tangent should not be confused. The given equation of the parabola which is of the form ${{y}^{2}}=4ax$ is open towards the positive x – axis and a parabola is a curve. A tangent on the other hand is a line that touches the curve at a single point and does not intersect at any other point even if the line is extended. Also, the equation of the parabola and the equation of a tangent to a curve should be correctly written and the values should also be correctly substituted. In the above solution, we started with a particular solution for tangents passing through the point P and then in the end, we generalized the obtained expression in terms of ‘x’ and ‘y’.