Question

Find $m$ if the line $y=mx-2$ intersects $y=x^2$ in just one point.

Answer 1

Step 1: Finding an equation for $m$.

Given,

$y=mx-2$ intersects $y=x^2$ at just one point.

Equating the two equations

When the line and the curve intersect, both will have a common solution and so we have: $y=y$.

So, the equation becomes,

$mx-2=x^2$

$\Rightarrow x^2-mx+2=0$

Step 2: Calculating the value of $m$:

When they have just one point of intersection, this means that the above quadratic equation has a distinctive solution and so,

$b^2=4ac$ where, $a=1,b=-m,c=2$.

So, we have

${(-m)}^2=4\times 1\times 2$

$\Rightarrow m^2=8$

Taking the square root on both sides we get,

$$\begin{gathered} m=\pm \sqrt{8} \\ \Rightarrow m=\pm 2\sqrt{2} \end{gathered}$$

Hence, the value of $m$ if the line$y=mx-2$ intersects $y=x^2$ in just one point is $m=\pm 2\sqrt{2}$.