Question

Find the condition on \[a\] and \[b\] so that the two tangents drawn to the parabola \[y^{2} = 4ax\] from a point are normal to the parabola \[x^2 = 4by\].
A) \[a^{2} > 8b^{2}\]
B) \[4a^{2} > b^{2}\]
C) \[a^{2} > 4b^{2}\]
D) \[8a^{2} > b^{2}\]

Answer 1

Hint: The equation of tangent to the parabola \[y^{2} = 4ax\] is given by \[y = mx+\dfrac{a}{m}\]. Here \[m\] is the slope of the line.
The equation of a line passing through the point \[(x_{1},y_{1})\] having slope \[m\] is given as,
\[y-y_{1} = m(x-x_{1})\].

Complete step-by-step answer:
Given the equation of parabolas are \[y^{2} = 4ax\] and \[x^2 = 4by\].
Consider the parabola \[x^2 = 4by\].
Any point on this parabola has the coordinates given by \[(2bt,bt^{2})\].
Differentiate the equation on both sides with respect to \[x\] to find the slope of the tangent to the parabola.
\[\begin{align*}2x &= 4b\dfrac{dy}{dx}\\ \dfrac{dy}{dx} &= \dfrac{x}{2b}\end{align*}\]
Since the normal is perpendicular to the tangent at a point, slope of the normal is given as,
\[\begin{align*}m_{N} &= -\dfrac{1}{dy/dx}\\ &= -\dfrac{2b}{x}\end{align*}\]
Thus the slope of the normal at the point \[(2bt,bt^{2})\] is \[-\dfrac{2b}{2bt} = -\dfrac{1}{t}\].
And the equation of the normal at that point is given using the formula \[y-y_{1} = m(x-x_{1})\] as,
\[\begin{align*}(y-bt^{2}) &= -\dfrac{1}{t}(x-2bt)\\ y &= -\dfrac{x}{t}+2b+bt^{2}\end{align*}\]
Since the tangents drawn to the parabola \[y^{2} = 4ax\] from a point are normal to the parabola \[x^{2} = 4by\], the equation \[y = -\dfrac{x}{t}+2b+bt^{2}\] is also the equation of the tangent.
But the equation of the tangent drawn to the parabola \[y^{2} = 4ax\] is given using the formula \[y = mx+\dfrac{a}{m}\].
This implies,
\[\begin{align*}2b+bt^{2} &= \dfrac{a}{-1/t}\\ bt^{2}-at+2b &= 0\end{align*}\]
The above quadratic equation has real roots and so its discriminant is greater than 0, i.e.,
\[\begin{align*}a^{2}-4(2b)\cdot b &> 0\\ a^{2}-8b^{2} &> 0\\ a^{2} &> 8b^{2}\end{align*}\]

Hence option (A) is correct.

Note: The condition can also be determined by determining the equation of the tangent line to the parabola ${y^2} = 4ax$ and the equation of normal to the parabola ${x^2} = 4by$ and then equating both the equations.