Question

If the two parabolas $y^2=4x$ and $y^2=(x-k)$ have a common normal other than the $x$-axis, then $k$ can be equal to

• A. $0$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is A $0$

Given,

$y^2=4x$

$4x=y^2$

$x=\frac{y^2}{4}$

Use the vertex form:

$x=a(y-k{)}^2+h$

to determine the values of a, h and k.

$a=\frac{1}{4},k=0,h=0$

Since the value of a is positive, the parabola opens right.

Axis of symmetry: $x=0$

Since parabolas have a common normal, axis of symmetry of prarabola $y^2=(x-k)$ also must be $x=0$.

So:

$x-k=0$

$0-k=0$

Add $k$ to both sides

$0-k+k=0+k$

$0=k$

$k=0$

Equation of parabola:

$y^2=(x-k)$

$y^2=(x-0)$

$y^2=x$