Question

What is parabola ? Write all the concepts, properties and formula involved in this chapter.Also,write equation of tangents and normal of the parabola. JEE ADVANCED PARABOLA​

Answer 1

Parabola

Parabola is a locus point whose distance from a fixed point and from a fixed line is constant.The fixed point is called focus and the fixed line is called directrix.

Consider the parabola y² = 4ax

Terminology related to parabola y² = 4ax

Axis

The line perpendicular to the directrix of the parabola and passing through its focus is called axis of the parabola.The equation of axis is given by y = 0.

Vertex

The mid-point of the focus and point of intersection of directrix and axis is called vertex of the parabola.The vertex of this parabola is (0,0)

The directrix of this parabola is the line x = - a .

The focus of the parabola has the co-ordinate (a,0).

Latus-rectum

Latus-rectum is a line perpendicular to axis of the parabola and passing through focus.In the given figure, please note that the end point of the latus-rectum is (a,2a) and (a,-2a) and therefore,its length is given by 4a.

It is a very important term.So,let us understand it with the help of some examples :-

Que:- What is the length of the latus-rectum of the parabola y² = 20x ?

Soln:- We know, the length of the latus-rectum of the parabola y² = 4ax is 4a. Hence, for y² = 20x ,it is 20

The value of a of this parabola is 5.

The equation of directrix is x = - a ,that means x = - 5.

The co-ordinate of focus is (5,0).

But, sometimes the parabola given will not be y² = 4ax, this can be like (y - 6)² = 8(x - 5)

So,let us understand how to deal with such problems.

Vertex

Vertex is given by x = 0 and y = 0.

Therefore,for this parabola ,it is x - 5 = 0 and y - 6 = 0. Hence, giving ,x = 5 and y = 6.

The co-ordinate of the vertex is (5,6).

Value of a

Given , equation is (y - 6)² = 8(x - 5). Let us see it as Y² = 4aX. Where, Y = y - 6 and X = x - 5 giving 4a = 8.

Thus, a = 2 for this parabola.

Length of latus-rectum

Length of latus-rectum is 4a.Thus,it is 8.

Directrix

Equation of directrix is X = - a. Thus, it is x - 5 = - 2

=> x = - 7 is the directrix of the parabola.

Axis

Equation of axis is given by Y = 0. Therefore, giving y - 6 = 0.

=> y = 6 is the axis of the parabola.

Parametric form of a parabola :-

The parametric form of a parabola is given by x = at² and y = 2at. This emplies that we can take any point on the parabola as (at²,2at).

Actually, what happens is simple. This co-ordinate satisfied the equation of parabola at any value of t.

proof :- LHS = y² = ( 2at)² = 4a²t²

RHS = 4ax = 4a × at² = 4a²t²

Thus, LHS = RHS ,at any value of t.

Tangent

The tangent of any equation can be simply found out by substituting :-

y² = y$y_1$

x² = x$x_1$

y = $\dfrac{y\:+\:y_1}{2}$

x = $\dfrac{x\:+\:x_1}{2}$

xy = $\dfrac{xy_1\:+\:yx_1}{2}$

Thus, the tangent to the parabola y² = 4ax is given by :-

y$y_1 \:=\:4a \dfrac{x\:+\:x_1}{2}$

=> y$y_1 \:=\:2a (x\:+\:x_1)$ is the equation of tangent of parabola in point-form.

By substituting x = at² and y = 2at in above equation , we get ty = x + at². It is known as parametric form of a parabola.

Equation of tangent in slope form :-

$y\:=\:mx\:+\: \dfrac{a}{m}$

Director circle

Director circle of a curve is a locus of the points from which perpendicular tangents can be drawn to the curve.

• The director circle of the parabola is its diretrix

What does it mean ?

It implies that if tangents are drawn from any point on the directrix to the parabola,they will be perpendicular.

Formulas :-

• Point of intersection of two tangents of parabola is given by $(at_1t_2,a(t_1+t_2))$

• Equation of normal in slope form :- y = mx - 2am - am³. Thus,3 tangents can be drawn to the parabola from a given point.This point is called co-normal point.

• If normal at any point $t_1$ to the parabola intersects the parabola at any point $t_2$ again ,then $t_2 \:=\: - \: t_1 \: - \: \dfrac{2}{t_1}$

• The length of latus-rectum if the parabola y = ax² + bx + c is 1/a.

• The length of latus-rectum of the parabola x = by² + cy + d is 1/b.

Properties

• The portion of any tangent between point of contact and directrix subtend 90° at focus.

• The perpendicular drawn to the tangent from focus meets the tangent at tangent at the vertex.

What if x = 0 is the axis of the parabola .

• Then equation is x² = 4ay

• Focus is (0,a).

• Length of latus-rectum is still 4a.

• Directrix is y = -a.

• Equation of tangent in parametric form is tx = y + at².

• Equation of tangent in slope form is y = mx - am².

Some General questions :-

Que:- For what value of c , the line y = 2x + c is a tangent to the parabola y² = 16x.

Hint :- m = 2 , a = 4 . The line will be tangent if c = a/m. (Remember slope form).

Que:- Find the equation of tangent in point form of the parabola is x² = 15y at (0,0).

Hint:- Remember what we have to substitute

Answer 2

Parabola :-

Parabola is a locus point whose distance from a fixed point and from a fixed line is constant.The fixed point is called focus and the fixed line is called directrix.