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Define parabola and with usual notation Prove that y2 = 4ax geometrically

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Definition of Parabola and other forms of parabola Parabola is a conic section whose eccentricity is 

1. That is, Parabola is the locus of the point which moves such that its distance from the fixed point (called the focus) is equal to its distance from the fixed line (called the directrix) We shall derive the equation of the parabola in its standard form.

Theorem: The equation of a Parabola with proper choice of coordinates axes is y2 = 4ax 

Proof: Let S be the focus and the line l be the directrix. 

Draw SZ through S and perpendicular to the directrix. 

Let O be the midpoint of the line segment SZ. 

Let SZ = 2a (a > 0) ⇒ OZ = OS = a 

We shall choose 0 as the origin and the line ZOS as x axis. 

The line YOY’ through O, perpendicular to the x-axis will be yaxis. 

With this choice of co-ordinate axes we have S(a, o) and Z(-a, 0) 

The equation of the directrix is l is x = -a 

Let P(x,y) be any point on the Parabola. 

Then by the definition of the Distance of P 

from S = Distance of P from the line ‘l’.

|SP| = |PM| 

SP2 = ZN2 since PM = ZN 

SP2 = (OZ + ON)2 

(x – a)2 + y2 = (a + x)2 (|OZ| = a, |ON| = x) 

x2 – 2ax + a2 + y2 = a2 + 2ax + x2 y2 = 4ax 

which is the equation of the parabola.

The equation y2 = 4ax is also called standard form of the equation of the parabola. 

Shape of the Parabola y2 = 4ax 

We shall note few observations from the equation y2 = 4ax, 

which will help us to trace the curve parabola. 

1. If y is replaced by -y in the equation remains same, 

i.e., (-y)2 = 4ax → y2 = 4ax. 

This shows that if (x, y) is any point on the curve y 2 = 4ax, then (x, -y) is also a point on the curve. 

Thus, the curve is symmetric about the x – axis, 

i.e., the shape of the curve above the x axis is the mirror image (about the x axis) of the shape of the curve below the x – axis. 

2. If x < 0, then y will be a negative quantity (note that a > 0) and therefore y2 = 4ax will have no real solution for y. This shows that no part of the curve lies to the left side of the x-axis. 

3. If y =0, the only value of x we get is zero. Thus the curve cuts the x axis at the origin(0,0) 

4. If x = 0, we get y2 = 0, which gives y = 0. Thus the curve cuts the y-axis at the origin and further the y-axis meets the curve only at the origin. That is, y – axis is a tangent to the curve at the origin. 

5. For any point P(x, y) on the parabola we have 

y2 = 4ax ⇒ y = ±2\(\sqrt{ax}\) 

y = 2\(\sqrt{ax}\) and y = -2\(\sqrt{ax}\) 

This shows that, as x increases from 0 to ∞, y also increases from 0 to ∞ or y decreases from 0 to -∞. Thus, the two branches of the parabola, laying on opposite sides of the x – axis, will extend to infinity towards the positive directions of the x – axis. 

From the above discussion and by plotting few points, whose coordinates satisfy y2 = 4ax, it is found the shape of the parabola is as shown in the following figure.

The origin 0 is called the Vertex of the parabola, y2 = 4ax, it is also denoted by V(0,0). 

The line ZSX is called the axis of the parabola and its equation is y = 0 

The focus S (a,0) and the equation of the directrix is x – a i.e., X + a = 0 

They – axis is called the tangent at the vertex – its equation is x = 0. 

Note: 

The distance between the vertex and the focus is the distance between the vertex and the directrix is equal to a and the distance between the directrix and the focus is 2 a. 

Definition: The chord passing through the focus and perpendicular to the axis of the parabola is called the latus rectum of the parabola.

In the figure LSL1 is the latus rectum. 

The points L and L1 on the parabola are called end points of the latus rectum, the length is called the length of the latus rectum. 

Clearly, the x – coordinates of L and L’ is a because OS = a. 

To find the corresponding coordinates, we shall put x = a in y2 = 4ax, we get y = ±2a. 

Thus, y – coordinates of L and L1 respectively. L = (a, 2a) and L1 = (a, -2a) 

∴ consider |LL’| = \(\sqrt{(a - a)^2 + (2a + 2a)^2} \)= √(4a)2 = 4a

Thus, the length of the latus rectum, LL1 = 4a.

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