Question

From an external point P tangents are drawn to the parabola; find the equation to the locus of P when these tangents make angles ${\theta }_1$ and ${\theta }_2$ with the axis, such that

$ta{n}^2{\theta }_1+ta{n}^2{\theta }_2$ is constant $(=\lambda )$.

Answer 1

Let the point P be $(a{t}_1{t}_2,a(t_1+t_2\left)\right)$

The tangent equations are $t_{1}y=x+a{t}_1^2$ and $t_{2}y=x+a{t}_2^2$

Since the tangents make angles ${\theta }_1$ and ${\theta }_2$ with the axis, $\frac{1}{t_1}=\tan {\theta }_1$ and $\frac{1}{t_2}=\tan {\theta }_2$

According to the given condition, $\frac{1}{(t_1{)}^2}+\frac{1}{(t_2{)}^2}=\lambda$

$\Rightarrow {t_1}^2+{t_2}^2=\lambda {t_1}^2{t_2}^2$

Let $a{t}_1{t}_2=x$ and $a(t_1+t_2)=y$

$\therefore (t_1+t_2{)}^2-2t_1{t}_2=\lambda {t_1}^2{t_2}^2$

$\therefore {\left(\frac{y}{a}\right)}^2-\frac{2x}{a}=\lambda \times \frac{x^2}{a^2}$

$\therefore y^2-2ax=\lambda x^2$ is the required locus