Question

Find the number of distinct real tangent that can be drawn from $(0,-2)$ to the parabola $y^2=4x$. Also find the slope of tangents.

Answer 1

Two tangents are possible from point $(0,-2)$ to the parabola.

equation of tangent

$\therefore ypo=2(x+\alpha )$

$\Rightarrow -2\beta =2\alpha$

$\Rightarrow \alpha =-\beta$

$\Rightarrow \because (\beta {)}^2=4\alpha (\because (\alpha ,\beta )alsoliesonthecurve)$

$\Rightarrow {\beta }^2=-4\beta$

$\Rightarrow {\beta }^2+4\beta =0$

$\Rightarrow \beta (\beta +4)=0\Rightarrow \beta 0,-4$

$\therefore \alpha =0,4$

$\therefore$ equation of tangent:-

$y\times 0=2(x+0)\Rightarrow x=0--\left(i\right)\Rightarrow slope=\infty$

$-4y=2(x+4)\Rightarrow x+2y+4=0\Rightarrow slope=-\frac{1}{2}$