From an external point P tangents are drawn to the parabola y² = 4ax and these tangents makes angles $\mathrm{\theta }$₁ ,$\mathrm{\theta }$₂ with its axis, such that cot $\mathrm{\theta }$₁ + cot $\mathrm{\theta }$_2 is a constant ‘d’ then S.T all such “p” lie on horizontal line

If $\mathrm{y}=\mathrm{mx}+\mathrm{c}$ is a tangent to parabola ${\mathrm{y}}^2=4\mathrm{ax}\text{ then }\mathrm{c}=\frac{\mathrm{a}}{\mathrm{m}}$
Let p(x₁, y₁) be any point,

 this line passes through P(x₁, y₁) then
${\mathrm{y}}_1={\mathrm{mx}}_1+\frac{\mathrm{a}}{\mathrm{m}}\Rightarrow {\mathrm{m}}^2{\mathrm{x}}_1-{\mathrm{my}}_1+\mathrm{a}=0\to \text{ (1) }$
Let m₁,m₂ are two roots of (1)
${\mathrm{m}}_1+{\mathrm{m}}_2=\frac{{\mathrm{y}}_1}{{\mathrm{x}}_1};{\mathrm{m}}_1{\mathrm{m}}_2=\frac{\mathrm{a}}{{\mathrm{x}}_1}$
Given $\cot {\mathrm{\theta }}_1+\cot {\mathrm{\theta }}_2=\mathrm{d}$
$\begin{array}{l}\frac{1}{\tan {\mathrm{\theta }}_1}+\frac{1}{\tan {\mathrm{\theta }}_2}=\mathrm{d}\\ \tan {\mathrm{\theta }}_1+\tan {\mathrm{\theta }}_2=\mathrm{d}\left(\tan {\mathrm{\theta }}_1\cdot \tan {\mathrm{\theta }}_2\right)\\ {\mathrm{m}}_1+{\mathrm{m}}_2=\mathrm{d}\left({\mathrm{m}}_1,{\mathrm{m}}_2\right)\Rightarrow \frac{{\mathrm{y}}_1}{{\mathrm{x}}_1}=\mathrm{d}\left(\frac{\mathrm{a}}{{\mathrm{x}}_1}\right)\\ \Rightarrow \frac{{\mathrm{y}}_1}{{\mathrm{x}}_1}=\mathrm{d}\left(\frac{\mathrm{a}}{{\mathrm{x}}_1}\right)\\ \Rightarrow {\mathrm{y}}_1=\mathrm{a}\cdot \mathrm{d}\end{array}$
$\text{ Locus of }\mathrm{P}\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)\text{ is }\mathrm{y}=\mathrm{ad}\text{ is a horizontal line }$