Important Parabola Formulas for JEE Main and Advanced

Parabola is an important topic for JEE Main and JEE Advanced. The parabola formulas given here will help students not only to strengthen their foundation in the topic but also to solve different types of problems easily. 

Parabola Formulas

1. Equation of standard parabola:

The equation of the parabola with focus at (a,0) a>0 and directrix x = -a is y2 = 4ax. Vertex is (0,0) and axis is y = 0. 

Length of latus rectum = 4a, ends of the latus rectum are L(a,2a), and L’(a,-2a).

2. Parametric representation:

x = at2 and y = 2at

3. Position of a point relative to a parabola:

The point (x1,y1) lies outside, on or inside the parabola y = 4ax according as y12-4ax1> , = or < 0

4. Line and a parabola:

Length of the chord intercepted by the parabola y2 = 4ax on the line y = mx+c is;

(4m2)a(1+m2)(amc)\left ( \frac{4}{m^{2}} \right )\sqrt{a(1+m^{2})(a-mc)}

5. Tangents to the parabola y2 = 4ax: T = 0

y = mx + am\frac{a}{m} where m ≠ 0 is the tangent to the parabola y2 = 4ax at (am2,2am)\left ( \frac{a}{m^{2}} , \frac{2a}{m}\right )

6. Normals to the parabola y2 = 4ax:

y-y1 = y12a(xx1)\frac{-y_{1}}{2a}(x-x_{1}) at (x1,y1) .

y = mx-2am-am3 at (am2, -2am)

y+tx = 2at+at3 at (at2, 2at)

7. Pair of tangents:

Let y2 = 4ax be the equation of a parabola and (x1, y1) an external point P. Then, equation of the tangents is given by

SS1 = T2, where S = y2-4ax, S1 = y12-4ax1, T = yy1 – 2a(x+x1).

8. Chord of contact:

Equation of the chord of contact of the tangents drawn from a point (x1, y1) to the parabola y2 = 4ax is T = 0, i.e. yy1 – 2a(x + x1) = 0. 

9. A chord with a given middle point:

The equation of the chord of the parabola y2 = 4ax with midpoint (x1, y1) is T = S1. 

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