Find the radius

Question

Consider the parabola $y=x^2$ and a circle which is tangent to the parabola at the points $(1,1)$ and $(-1,1)$.Find the radius of circle.

My try:I write the general equation of circle

$(x-h)^2+(y-k)^2=r^2$

and substitute the points $(1,1)$ and $(-1,1)$ in the equation of circle,i find $h=0$.

Now further what i should do to find r?

Answer 1

The tangent line to the parabola at $(1,1)$ has slope $2$. This is also the tangent line to the circle, so the radius from the center to $(1,1)$, which is perpendicular to the tangent line, has slope $-1/2$. This line intersects the $y$-axis at $(0,3/2)$, so this is your center. To find the radius just take the distance to $(1,1)$.

Answer 2

Substitute $(1,1),(-1,1)$ in $(x-h)^2+(y-k)^2=r^2$

you will get

$h^2+k^2-2h-2k+2=r^2$ and $h^2+k^2+2h-2k+2=r^2$

From here you get $h=0$, substitute this in one of the above to obtain

$k^2-2k+2=r^2$

$m_1$=slope of tangent line at $(1,1)=2$ (here use the $y=x^2$)

Where $m_2$=the slope of radial line at $(1,1)= 1-k$ (here use the equation of circle)

since $m_1m_2=-1$, so we have $k=\frac{3}{2}$

Substitute $k=\frac{3}{2}$ in

$$k^2-2k+2=r^2$$

Finally you will get the answer $r=\frac{\sqrt{5}}{2}$