skills

To complete this question, you will need to be able to determine the algebraic equation of a curve from a geometric description.

Question

Let $L$ be the line $x=0$. Let $A$ be the point with coordinates $(4,0)$. The curve $C$ is defined at the set of points $P$ such that the horizontal distance from $L$ is equal to the distance from the point $A$. Show that the curve $C$ is a parabola, writing its equation in the form $x = ay^2 + by + c$, where $a,b,c \in \mathbb{R}$.

answer --- press button to toggle display

$$x = \frac{1}{8}y^2 + 2 $$

solution --- press button to toggle display

By drawing a diagram, we can more clearly identify the curve $C$.

The horizontal distance from the line $L$ is also the minimum distance from the line $L$. Consequently, we are concerned with the set of points satisfying the equation $$ |LP| = |AP| $$ Since the length $LP$ is the horizontal distance from the x-axis, $|LP| = x$. The distance from $A(4,0)$ to the point $P(x,y)$ is $$ |AP| = \sqrt{(4-x)^2 +y^2} $$ We therefore have arrive at the equation $$ \begin{array}{ccc} |LP| &=& |AP| \\ x &=& \sqrt{(4-x)^2 +y^2} \\ x^2 &=& 16 - 8x + x^2 + y^2 \\ x &=& \frac{1}{8}y^2 + 2 \\ \end{array} $$