Question

The tangent to (at², 2at) is perpendicular to X-axis at......,Select correct option from the given options.
(a) (4a,4a)
(b) (a,2a)
(c) (0,0)
(d) (a,-2a)

Answer 1

(at² , 2at) is parametric equation of parabola y² = 4ax
slope of tangent of parabola at (at²,2at) = $\frac{dy}{dx}|_{(at^2,2at)}$

y² = 4ax
differentiate both sides with respect to x,
2y dy/dx = 4a

so, dy/dx = 2a/y

at (at²,2at) , slope of tangent = 2a/2at = 1/t

now, equation of tangent : $(y-2at)=\frac{1}{t}(x-at^2)$
tangent is perpendicular to X-axis , means slope of tangent must be infinity.
means, 1/t = ∞

this is possible in case of y² = 4ax only when tangent is Y-axis e.g., at origin Y-axis touch the parabola y² = 4ax and also Y-axis is perpendicular to X-axis.

so, the correct choice should be (c) (0,0)

Answer 2

Dear Student:

slope =m = ∞

(at², 2at) tangent perpendicular to x-axis

x=0 equation of tangent.

Option c is correct.

See the attachment.