Question

Find the angle between two tangents drawn to the parabola $y=x^2$ from the point $(0,-2)$.

Answer 1

Parabola: $y=x^2$

Tangent are drawn from $(0,-2),$

$S{S}_1=T^2$

$=>(x^2-y)(0-(-2\left)\right)=[0-\frac{(y-2)}{2}{]}^2$

$=>4(2x^2-2y)=y^2+4-4y$

$=>{8x}^2-4y-y^2-4=0$

$=>8x^2-(y^2+4y+4)=0$

$=>8x^2-{(y+2)}^2=0$

$=>(2\sqrt{2}x-y-2)(2\sqrt{2}x+y+2)=0$

Equation of tangents, $T_1:2\sqrt{2}x-y-2=0$and$T_2:2\sqrt{2}x+y+2=0$

$=>$Slope of$T_1=m_1=2\sqrt{2}$

$=>$Slope of$T_2=m_2=-2\sqrt{2}$

Now angle between these tangents, $\alpha ={\tan }^{-1}\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$

$=>\alpha ={\tan }^{-1}\left|\frac{2\sqrt{2}+2\sqrt{2}}{1-8}\right|$

$=>\alpha ={\tan }^{-1}\frac{4\sqrt{2}}{7}.$