Question

two tangents are drawn from the point -2,1 to the parabola y2 =4x if theta is angle between tangent then tan theta

Answer 1

$\Large\bf{\color{cyan}GiVeN,}$

Two tangents are drawn from the point (-2 , 1) to the parabola $\bf{y^2\:=\:4x}$.

☆ Equation of parabola,

$\orange\bigstar\:\:\bf\blue{y^2\:=\:4ax}$

☆ According to the question,

a = 1

☆ Equation of tangent,

$\green\bigstar\:\:\bf\purple{y\:=\:mx\:+\:\dfrac{a}{m}\:}$

$\longmapsto\:\:\bf{y\:=\:mx\:+\:\dfrac{1}{m}\:}$

$\bf\red{Here,}$

x = -2

y = 1

$\longmapsto\:\:\bf{1\:=\:-2m\:+\:\dfrac{1}{m}\:}$

$\longmapsto\:\:\bf{m\:=\:-2m^2\:+\:1\:}$

$\longmapsto\:\:\bf{2m^2\:+\:m\:-\:1\:=\:0\:}$

$\longmapsto\:\:\bf{2m^2\:+\:2m\:-\:m\:-\:1\:=\:0\:}$

$\longmapsto\:\:\bf{2m\:(m\:+\:1)\:-\:1\:(m\:+\:1)\:=\:0\:}$

$\longmapsto\:\:\bf{(2m\:-\:1)\:(m\:+\:1)\:=\:0\:}$

$\longmapsto\:\:\bf{m\:=\:\dfrac{1}{2}\:~~~or~~~\:m\:=\:-1\:}$

$\bf\pink{Let,}$

$\bf{m_1\:=\:\dfrac{1}{2}}$

$\bf{m_2\:=\:-1}$

$\purple\bigstar\:\:\bf{\color{peru}\tan{\theta}\:=\:\mid\:{\dfrac{m_1\:-\:m_2}{1\:+\:m_1\:m_2}}\:\mid}$

$:\implies\:\:\bf{\tan{\theta}\:=\:\mid\:{\dfrac{0.5\:-\:(-1)}{1\:+\:0.5\:(-1)}}\:\mid}$

$:\implies\:\:\bf{\tan{\theta}\:=\:\mid\:{\dfrac{1.5}{1\:-\:0.5}}\:\mid}$

$:\implies\:\:\bf{\tan{\theta}\:=\:\mid\:{\dfrac{1.5}{0.5}}\:\mid}$

$:\implies\:\:\bf\orange{\tan{\theta}\:=\:3\:}$