Review question

# How many common tangents can two parabolas have? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R7750

## Question

Let $P$ be the point on the curve $y=ax^2+bx+c$ (where $a$ is non-zero) at which the gradient is $m$. Show that the equation of the tangent at $P$ is $y-mx=c-\frac{(m-b)^2}{4a}.$

Show that the curves $y=a_1x^2+b_1x+c_1$ and $y=a_2x^2+b_2x+c_2$ (where $a_1$ and $a_2$ are non-zero) have a common tangent with gradient $m$ if and only if $(a_2-a_1)m^2+2(a_1b_2-a_2b_1)m+4a_1a_2(c_2-c_1)+a_2b_1^2-a_1b_2^2=0.$

Show that, in the case $a_1\neq a_2$, the two curves have exactly one common tangent if and only if they touch each other. In the case $a_1=a_2$, find a necessary and sufficient condition for the two curves to have exactly one common tangent.