Find the equations of the common tangents to the parabola $y^2=15x$ and the circle $x^2+y^2=16$.

Question

The text says:

Find the equations of the common tangents to the parabola $y^2=15x$ and the circle $x^2+y^2=16$.

I tried the approach of the discriminant and also one using the distance from a line but both didn't work for me. A previous exercise asked me to demonstrate that the line $y=mx+\frac{15}{4m}$ is a tangent to the parabola for every value of $m$. A suggestion in the text says I can use this result also to find the common tangent.

Answer 1

Let a common tangent touch the circle at $\displaystyle (a,b)$ and the parabola at $\displaystyle (\alpha, \beta)$, and let it have the equation $\displaystyle y = mx + c$.

Now proceed systematically and list what you know, writing equations along the way.

1) $\displaystyle (a,b)$ satisfies the equation of the tangent. Hence $\displaystyle b = ma + c$

2) $\displaystyle (a,b)$ satisfies the equation of the circle. Hence $\displaystyle a^2 + b^2 = 16$

3) $\displaystyle (\alpha,\beta)$ satisfies the equation of the tangent. Hence $\displaystyle \beta = m\alpha + c$

4) $\displaystyle (\alpha,\beta)$ satisfies the equation of the parabola. Hence $\displaystyle \beta^2 = 15\alpha$

5) The tangent to the circle at $\displaystyle (a,b)$ is equal to $\displaystyle m$. By implicit differentiation, you know that $\displaystyle 2x + 2yy' = 0$ defines the tangent to the circle at an arbitrary point, so you get $\displaystyle m = -\frac ab$

6) The tangent to the parabola at $\displaystyle (\alpha, \beta)$ is equal to $\displaystyle m$. By implicit differentiation, you know that $\displaystyle 2yy' = 15$ defines the tangent to the parabola at an arbitrary point, so you get $\displaystyle m = \frac {15}{2\beta}$

You now have a system of $\displaystyle 6$ equations in $\displaystyle 6$ unknowns. Proceed to solve them, remembering that you're trying to get values for $m$ and $c$ and eliminating the other variables systematically.

You should finally get $\displaystyle m = \pm \frac 34$ and $\displaystyle c = \pm 5$, giving you the equations of the common tangents as $\displaystyle y = \pm (\frac 34x + 5)$

Here is a graphical representation.

Answer 2

HINT:

To tackle the second part together it is better to use symbols throughout.

Circle tangent is

$$ x x_1 + y y_1 = a^2, slope= \frac {x_1}{y_1} $$

Parabola is

$$ y^2 = 4 f x $$

Differentiate the above for parabola

$$ y^{\prime}_{par} = 2 f/y_{par} $$

Equating common slopes we get

$$ x_{par} = 2f,\, y_{par} = 2 \sqrt2\, f $$

Verify the above. Can you now take it further (to find equation to a line passing through above point and given slope) ?