Name: Normals to the Parabolas
Uploaded: 2019-07-19
Duration: 9 min 41 s
Description: This video will give a clear understanding of how to write and learn the concept of normal to the parabola in the parametric form. Don't miss it.

Question

Prove that the equation of the parabola whose focus is

(0, 0)

and tangent at the vertex is

x - y + 1 = 0

is

x^{2} + y^{2} + 2 x y - 4 x + 4 y - 4 = 0

.

Accepted Answer

Given: tangent at the vertex is $x - y + 1 = 0 . . . . . (i)$ and focus is $(0, 0)$

As we know axis of the parabola is perpendicular to the tangent at the vertex.

So, the slope of the axis of the parabola $= - \frac{1}{Slope of tangent at vertex} = - 1$

Since, the axis of the parabola is passing through the focus $(0, 0)$ .

$∴$ Equation of the axis of parabola is $y - 0 = - 1 (x - 0)$

$\Rightarrow y + x = 0 . . . . . (i i)$

On solving equation $(i)$ and equation $(i i)$ , we get

$x = - \frac{1}{2}$ and $y = \frac{1}{2}$

$∴$ Coordinates of the vertex are $(- \frac{1}{2}, \frac{1}{2})$

Let the coordinates of the point $x - y + 1 = 0$ on directrix be $(a, b)$ .

$x^{2} + y^{2} + 2 x y - 4 x + 4 y - 4 = 0$ and $\frac{b + 0}{2} = \frac{1}{2}$

$\Rightarrow a = - 1$ and $b = 1$

Now, the slope of the directrix $=$ slope of the tangent at the vertex ( $∵$ both lines are parallel)

So, the equation of the directrix is $y - 1 = 1 (x + 1)$

$\Rightarrow y - x - 2 = 0$

Let $P$ $(x, y)$ be any point on the parabola and the perpendicular distance from point $P$ to the directrix be $P M$ .

So, by definition of parabola $S P = P M$

$\Rightarrow \sqrt{{(x - 0)}^{2} + {(y - 0)}^{2}} = \frac{|y - x - 2|}{\sqrt{{(1)}^{2} + {(- 1)}^{2}}}$

$\Rightarrow \sqrt{x^{2} + y^{2}} = \frac{|y - x - 2|}{\sqrt{2}}$

$\Rightarrow 2 (x^{2} + y^{2}) = {(y - x - 2)}^{2}$

$\Rightarrow 2 x^{2} + 2 y^{2} = y^{2} + x^{2} + 4 - 2 x y + 4 x - 4 y$

$\Rightarrow x^{2} + y^{2} + 2 x y - 4 x + 4 y - 4 = 0$

Prove that the equation of the parabola whose focus is $(0, 0)$ and tangent at the vertex is $x - y + 1 = 0$ is $x^{2} + y^{2} + 2 x y - 4 x + 4 y - 4 = 0$ .

Important Questions on Conic Section

Learn and Prepare for any exam you want !

Prove that the equation of the parabola whose focus is 0,0 and tangent at the vertex is x-y+1=0 is x2+y2+2xy-4x+4y-4=0.

Important Questions on Conic Section

Learn and Prepare for any exam you want !

Prove that the equation of the parabola whose focus is $(0, 0)$ and tangent at the vertex is $x - y + 1 = 0$ is $x^{2} + y^{2} + 2 x y - 4 x + 4 y - 4 = 0$ .