Review question

# When does the normal to a parabola pass through this point? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6352

## Question

As shown in the diagram below: $C$ is the parabola with equation $y=x^2$; $P$ is the point $(0,1)$; $Q$ is the point $(a,a^2)$ on $C$; $L$ is the normal to $C$ which passes through $Q$.

1. Find the equation of $L$.

2. For what values of $a$ does $L$ pass through $P$?

3. Determine $\big |QP\big |^2$ as a function of $a$, where $\big |QP\big |$ denotes the distance from $P$ to $Q$.

4. Find a point $R$, in the $xy$-plane but not on $C$, such that $\big |RQ\big |$ is smallest for a unique value of $a$. Briefly justify your answer.