Review question

# What is implied if these two parabolas touch exactly once? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R5739

## Question

It is given that the two curves $y=4-x^2 \qquad \text{and} \qquad mx=k-y^2,$ where $m>0$, touch exactly once.

1. In each of the following four cases, sketch the two curves on a single diagram, noting the coordinates of any intersections with the axes:

1. $k<0$;

2. $0<k<16$, $k/m<2$;

3. $k>16$, $k/m>2$;

4. $k>16$, $k/m<2$.

2. Now set $m=12$. Show that the $x$-coordinate of any point at which the two curves meet satisfies $x^4-8x^2+12x+16-k=0.$ Let $a$ be a value of $x$ at the point where the curves touch. Show that $a$ satisfies $a^3-4a+3=0$ and hence find the three possible values of $a$.

Derive also the equation $k=-4a^2+9a+16.$ Which of the four sketches in part (i) arise?