Question

The $y$ intercept of the common tangent to the parabola ${y^2} = 32x$ and ${x^2} = 108y$ is?
A.$ - 18$
B.$ - 12$
C.\[ - 9\]
D.\[ - 6\]

Answer 1

Hint: First of all compare the given equation with the equation of the parabola ${y^2} = 4ax$ to find the value of $a$. Then write the equation of the tangent to the parabola ${y^2} = 4ax$. Since the tangent is common to the parabola ${x^2} = 108y$, substitute the equation of a line to form a quadratic equation in $x$.At last, put the discriminant of the equation equals to find the slope of the tangent common to parabola ${y^2} = 32x$ and ${x^2} = 108y$. Substitute 0 for $x$ in the formed equation of a tangent to find $y$ intercept.

Complete step by step answer:

Compare the given equation of parabola with the standard equation of ${y^2} = 4ax$

$
 \Rightarrow 32x = 4ax \\
 \Rightarrow 4a = 32 \\
 \Rightarrow a = 8 \\
$
The equation of the tangent to the parabola ${y^2} = 32x$ is given by,
\[y = mx + \dfrac{8}{m}{\text{ }}\left( 1 \right)\]
Also, the tangent meets the parabola, ${x^2} = 108y$
On substituting the value of $y = mx + \dfrac{8}{m}{\text{ }}$ in the equation of ${x^2} = 108y$, we get,
$
  \Rightarrow{x^2} = 108\left( {mx + \dfrac{8}{m}{\text{ }}} \right) \\
  \Rightarrow m{x^2} = 108{m^2}x + 864 \\
 \Rightarrow m{x^2} - 108{m^2}x - 864 = 0 \\
$
If the line $y = mx + \dfrac{8}{m}{\text{ }}$ is tangent to the parabola then roots of the equation $m{x^2} - 108{m^2}x - 864 = 0$must be equal.
Also, we can say that the discriminant of the equation must be 0.
Discriminant of an equation $a{x^2} + bx + c = 0$ is $D = {b^2} - 4ac$ .
On substituting the values of $a = m,b = - 108{m^2},c = - 864$ in the formula of discriminant, $D = {b^2} - 4ac$, we get,
$D = {\left( { - 108{m^2}} \right)^2} - 4\left( m \right)\left( { - 864} \right)$
Equate it to 0 to find the value of $m$.
$
 \Rightarrow {\left( { - 108{m^2}} \right)^2} - 4\left( m \right)\left( { - 864} \right) = 0 \\
  \Rightarrow 27{m^3} + 8 = 0 \\
 \Rightarrow {m^3} = - \dfrac{8}{{27}} \\
  \Rightarrow m = - \dfrac{2}{3} \\
 $
On substituting the value of $m$ in equation (1) we get,
$
  \Rightarrow y = \left( { - \dfrac{2}{3}} \right)x + \dfrac{8}{{ - \dfrac{2}{3}}} \\
  \Rightarrow y = \dfrac{{ - 2x - 36}}{3} \\
  \Rightarrow 3y = - 2x - 36 \\
  \Rightarrow 2x + 3y + 36 = 0 \\
$
Thus, the equation of tangent is, $2x + 3y + 36 = 0$
To find the $y$intercept of substitute 0 for $x$.
$
 \Rightarrow 2\left( 0 \right) + 3y + 36 = 0 \\
 \Rightarrow 3y = - 36 \\
 \Rightarrow y = - 12 \\
$
Hence, option B is the correct answer.

Note: The equation of the line in slope-intercept form is, \[y = mx + c\]. If the line touches the parabola, then the discriminant of the resulting quadratic equation is 0. We have to substitute $x$ as 0 to find the $y$ intercept.