Mathematics - Parabola Question with Solution | TestHub

Step 1: Write the equations of the normals

The equation of the normal in slope form to the parabola

$y^2=4ax$ is given by

$y=mx-2am-a{m}^3$.

For the parabola

$y^2=4c(x-b)$, which is a standard parabola shifted by $b$ along the x-axis, the equation of the normal is

$y=m(x-b)-2cm-c{m}^3$, which simplifies to

$y=mx-mb-2cm-c{m}^3$.

Step 2: Equate the equations for a common normal

For a common normal to exist, the two equations must represent the same line. This means their coefficients must be equal. Equating the constant terms (since the $mx$ terms are already the same):

$-2am-a{m}^3=-mb-2cm-c{m}^3$

Step 3: Rearrange and factor the equation

Rearrange the equation to one side:

$a{m}^3+c{m}^3+2am+mb+2cm=0$

Factor out $m$:

$m^3(a+c)+m(2a+b+2c)=0$

$m((a+c)m^2+(2a+b+2c))=0$

Step 4: Determine the condition for a real normal

The solution $m=0$ corresponds to the axis of symmetry (the x-axis, $y=0$), which is a common normal for all such parabolas. For a common normal *other* than the x-axis, we must have a real non-zero value for $m$. This requires the quadratic factor to be solvable for a real $m$:

$(a+c)m^2+(2a+b+2c)=0$

$m^2=-\frac{2a+b+2c}{a+c}$

For $m$ to be a real number, $m^2$ must be non-negative ($m^2>0$ as $m\ne 0$). Thus, we need:

$-\frac{2a+b+2c}{a+c}>0$

Since $a$ and $c$ are typically considered positive constants in this context, the denominator $(a+c)$ is positive. Therefore, the numerator must be negative:

$-(2a+b+2c)>0$

$2a+b+2c<0$

However, this leads to a contradiction if $a,b,c$ are positive.

Revisiting the derivation by equating coefficients more directly:

$m^2=\frac{b+2c-2a}{a-c}$

For $m^2>0$, the numerator and denominator must have the same sign. Assuming $a,c$ are distinct positive real numbers, if we assume $a>c$, then $a-c>0$, so we must have $b+2c-2a>0$, which gives $b>2a-2c$, or $b>2(a-c)$. This can be expressed as $\frac{b}{a-c}>2$.

**Answer:** The condition for the parabolas to have a common normal other than the x-axis is $\frac{b}{a-c}>2$ (or equivalently, $b>2(a-c)$, assuming $a,c$ are positive and $a>c$).