Name: Conic Sections: The Parabola
Uploaded: 2019-07-19
Duration: 9 min 43 s
Description: This video defines a parabola and explains how to graph a parabola in standard form, which will help you to solve questions in exam. Watch and learn.

Question

The normals are drawn from

(2 λ, 0)

to the parabola

y^{2} = 4 x

. Show that $\lambda$ must be greater than

1

. One normal is always the

x

-axis. Find

λ

for which the other two normals are perpendicular to each other.

Accepted Answer

The given parabola is $y^{2} = 4 x . . . (1)$

Compare it with standard parabola, $y^{2} = 4 a x$ , we get $a = 1$

The coordinates of the parabola is $(a t^{2}, 2 a t)$ i.e. $(t^{2}, 2 t)$

Differentiate $(1)$ w.r.t. $x$ , we get

$2 y \frac{d y}{d x} = 4 \Rightarrow \frac{d y}{d x} = \frac{2}{y}$

Thus, the slope of normal, $m = - \frac{d x}{d y} = - \frac{y}{2} = - t$

The equation of normal is $y - y_{1} = - \frac{d x}{d y} (x - x_{1})$

$\Rightarrow y - 2 t = - t (x - t^{2}) \Rightarrow y + t x - 2 t - t^{3} = 0 \Rightarrow y - m x + 2 m + m^{3} = 0 [∵ t = - m]$

Normal is drawn from $(2 λ, 0)$

$\Rightarrow 0 - 2 m λ + 2 m + m^{3} = 0 \Rightarrow m (m^{2} + 2 - 2 λ) = 0 \Rightarrow m = 0, m^{2} = 2 λ - 2$

which gives one value of $m$ as $0$ i.e. one normal on $x -$ axis but for three distinct normals, the roots of the above equation should be real and distinct

$\Rightarrow 2 λ - 2 > 0 \Rightarrow λ > 1$

Now, $m^{2} + 2 - 2 λ = 0$

Two normals are perpendicular if $m_{1} . m_{2} = - 1$

$\Rightarrow \frac{2 - 2 λ}{1} = - 1 \Rightarrow 2 λ = 3 \Rightarrow λ = \frac{3}{2}$

Hence, the other two normals are perpendicular for $λ = \frac{3}{2}$

The normals are drawn from $(2 λ, 0)$ to the parabola $y^{2} = 4 x$ . Show that $\lambda$ must be greater than $1$ . One normal is always the $x$ -axis. Find $λ$ for which the other two normals are perpendicular to each other.

Important Questions on Parabola

Learn and Prepare for any exam you want !

The normals are drawn from 2λ,0 to the parabola y2=4x. Show that $\lambda$ must be greater than 1. One normal is always the x-axis. Find λ for which the other two normals are perpendicular to each other.

Important Questions on Parabola

Learn and Prepare for any exam you want !

The normals are drawn from $(2 λ, 0)$ to the parabola $y^{2} = 4 x$ . Show that $\lambda$ must be greater than $1$ . One normal is always the $x$ -axis. Find $λ$ for which the other two normals are perpendicular to each other.