Question

If the tangents at two points of a parabola are at right angles, then show that they intersect at a point on the directrix.

Answer 1

We remind that the parabola is the locus of points that are equidistant from both focus and the directrix . We assume that the focus has coordinates $(x_0,y_0)$ and $y=y_1$ is an equation of directrix. The following equality has to be satisfied:

$|y-y_1|=\sqrt{(x-x_0)^2+(y-y_0)^2}$

From the latter, by direct calculations, we get:

$(y-y_1)^2=(x-x_0)^2+(y-y_0)^2\,\,\,\Longrightarrow\,\,\,\,y=\frac{(x-x_0)^2}{2(y_0-y_1)}$ .

We remind that the equation of the tangent line to the curve $y=f(x)$ (see e.g., https://en.wikipedia.org/wiki/Tangent) at point $\tilde{x}$ has the form:

$y=f(\tilde{x})+f'(\tilde{x})(x-\tilde{x})$

From this we receive that equation of the tangent line to parabola has the form:

$y=\frac{(\tilde{x}-x_0)^2}{2(y_0-y_1)}+\frac{\tilde{x}}{y_0-y_1}(x-\tilde{x})$

Suppose that we have two tangent lines at points $\tilde{x}_1$ and $\tilde{x}_2$. The tangent of the angle between those two lines is (see https://en.wikipedia.org/wiki/List_of_trigonometric_identities):

$tan\,\alpha=\frac{\frac{\tilde{x}_1-\tilde{x}_2}{y_0-y_1}}{1+\frac{\tilde{x}_2}{y_0-y_1}\cdot\frac{\tilde{x}_1}{y_0-y_1}}=\frac{\tilde{x}_1-\tilde{x}_2}{y_0-y_1+{\tilde{x}_2}\tilde{x}_1}$

The tangent tends to infinity for the angles close to 90⁰

This means, that $y_0-y_1+{\tilde{x}_2}\tilde{x}_1=0$. From the latter we found that $\tilde{x}_2=\frac{y_1-y_0}{\tilde{x}_1}$. In order to find the point, where two lines intersect, we have the equality:

$\frac{(\tilde{x}_1-x_0)^2}{2(y_0-y_1)}+\frac{\tilde{x}_1}{y_0-y_1}(x-\tilde{x}_1)=\frac{(\tilde{x}_2-x_0)^2}{2(y_0-y_1)}+\frac{\tilde{x}_2}{y_0-y_1}(x-\tilde{x}_2)$

After simplifications we receive:

$x=x_0+\frac{\tilde{x}_1+\tilde{x}_2}{2}$

We substitute the latter in equation for the tangent line and receive the y-coordinate of the point, where tangent lines intersect. We point out, that in general it is not equal to $y_1$