# Find the points of intersection of two perpendicular tangents to a parabola $y^2=4ax$.

Find the points of intersection of two perpendicular tangents to a parabola $y^2=4ax$.

Any point on the parabola is $(at^2,2at)$. Hence if they intersect at $(x_1,y_1)$ then $\dfrac{2at-y_1}{at^2-x_1}=\dfrac{1}{t}\implies at^2-ty_1+x_1=0$ since the slope at $(at^2,2at)=\dfrac{1}{t}$.

In order to the tangents to perpendicular $t_1=t_2$.

Let $$(x,y)=(at^{2},2at)$$, then the equation of the tangent at $$[t]$$ is

$$x-ty+at^{2}=0 \tag{1}$$

If tangents at $$[t]$$ and $$[t']$$ are perpendicular, $$tt'=-1$$

Then the equation of another tangent at $$[t']$$ is $$x+\frac{y}{t}+\frac{a}{t^{2}}=0$$

That is $$t^{2}x+ty+a=0 \tag{2}$$

$$(1)+(2)$$, $$(1+t^2)x+a(t^2+1)=0$$

$$\fbox{x=-a}$$

The locus is the directrix of the parabola which is the limiting case for director circle of an ellipse or a hyperbola.

Useful fact:

Equation of tangent for conics $$ax^2+2hxy+by^2+2gx+2fy+c=0$$ at the point $$(x',y')$$ is given by

$$ax'x+h(y'x+x'y)+by'y+g(x+x')+f(y+y')+c=0$$

From this, $$m^2x-my+a=0$$
If $m_1,m_2$ are the roots of the above equation, $m_1+m_2=?, m_1m_2=?$
If the angle between the tangents is $\theta$ $$\tan\theta=\left|\dfrac{m_1-m_2}{1+m_1m_2}\right|$$
Square both sides and use $$(m_1-m_2)^2=(m_1+m_2)^2-4m_1m_2$$
Here’s a somewhat different approach. Working in homogeneous coordinates, if we represent a non-degenerate conic with a matrix $C$, i.e., the conic is the set $\{\mathbf x \mid \mathbf x^TC\mathbf x=0 \}$, then every tangent line $\mathbf l$ to the parabola satisfies the dual conic equation $\mathbf l^TC^{-1}\mathbf l=0$. For this parabola, $$C = \begin{bmatrix}0&0&2a\\0&-1&0\\2a&0&0\end{bmatrix},$$ so by setting $\mathbf l = [\lambda:\mu:\tau]$ and solving for $\tau$, we find that its tangents are of the form $\mathbf l = \left[\lambda:\mu:\frac{a\mu^2}\lambda\right]$, $\lambda\ne0$. (If $\lambda=0$, the line is horizontal and can’t be tangent to the parabola.) The perpendicular tangent to this can be found by making the substitutions $\lambda\mapsto\mu$ and $\mu\mapsto-\lambda$ to get $\mathbf m = \left[\mu:-\lambda:\frac{a\lambda^2}\mu\right]$ ($\mu\ne0$), and the intersection of these two lines is $$\mathbf l\times\mathbf m = \left[a(\lambda^2+\mu^2) : a\left(\frac{\mu^3}\lambda-\frac{\lambda^3}\mu\right) : -(\lambda^2+\mu^2)\right]$$ which is $\left(-a, a\left(\frac\lambda\mu-\frac\mu\lambda\right)\right)$ in inhomogeneous Cartesian coordinates. The parameters $\lambda$ and $\mu$ can be chosen to produce any value whatsoever for the $y$-coordinate, so the locus of intersection is the line $x=-a$, the parabola’s directrix.