The equations of the tangent and normal to the parabola y2=4 a x at the point (a t2, 2 a t) are respectively

Question

The equations of the tangent and normal to the parabola y2=4 a x at the point (a t2, 2 a t) are respectively (A) t y=x+a t2 and y=-t x+2 a t+a t3 (B) t y=x-a t2 and y=t x-2 a t+a t3 (C) y=t x+2 a t+a t3 and t y=x+a t2 (D) y=-t x+2 a t+a t3 and t y=x+a t2

Tardigrade Admin · Accepted Answer

The equation of the given parabola is y2=4 a x ......(i) On differentiating w.r.t. x, we get 2 y (d y/d x)=4 a ⇒ (d y/d x)=(2 a/y) ∴ Slope of the tangent at (a t2, 2 a t) is ((d y/d x))(a t2, 2 a t)=(2 a/2 a t)=(1/t) Hence, the equation of the tangent at (a t2, 2 a t) is y-2 a t =(1/t)(x-a t2) ⇒ y t-2 a t2=x-a t2 ⇒ x-t y+a t2=0 Also, slope of the normal at (a t2, 2 a t)=(-1/ text Slope of tangent ) a t(a t2, 2 a t)=-t ∴ The equation of the normal at (a t2, 2 a t) is y-2 a t=-t(x-a t2) ⇒ t x+y-2 a t-a t3=0

Q. The equations of the tangent and normal to the parabola $y^{2} = 4 a x$ at the point $(a t^{2}, 2 a t)$ are respectively

$t y = x + a t^{2}$ and $y = - t x + 2 a t + a t^{3}$

$t y = x - a t^{2}$ and $y = t x - 2 a t + a t^{3}$

$y = t x + 2 a t + a t^{3}$ and $t y = x + a t^{2}$

$y = - t x + 2 a t + a t^{3}$ and $t y = x + a t^{2}$

Q. The equations of the tangent and normal to the parabola y2=4ax at the point (at2,2at) are respectively

ty=x+at2 and y=−tx+2at+at3

ty=x−at2 and y=tx−2at+at3

y=tx+2at+at3 and ty=x+at2

y=−tx+2at+at3 and ty=x+at2

Q. The equations of the tangent and normal to the parabola $y^{2} = 4 a x$ at the point $(a t^{2}, 2 a t)$ are respectively

$t y = x + a t^{2}$ and $y = - t x + 2 a t + a t^{3}$

$t y = x - a t^{2}$ and $y = t x - 2 a t + a t^{3}$

$y = t x + 2 a t + a t^{3}$ and $t y = x + a t^{2}$

$y = - t x + 2 a t + a t^{3}$ and $t y = x + a t^{2}$