## 1.Equations of tangent at a point on the parabola:

**(i) Point Form:** The equation of the tangent to a parabola *y*^{2} = 4*ax* at a point *P*(*x*_{1}, *y*_{1}) on the parabola is

*yy*_{1} = 2*a* (*x+x*_{1})

**Note:** The equation of the tangent to a parabola (*bx – ay*)^{2} + 2*gx* + 2*fy + d* = 0 at a point *P*(*x*_{1}, *y*_{1}) is obtained by replacing:

**(ii) Slope Form:** The equation of line which has a slope *m* and is tangent to the parabola *y*^{2} = 4*ax* is

**(ii) Condition of tangency:**

(b) A line *lx + my + n =* 0 is tangent to parabola *y*^{2} = 4*ax* if *ln = am*^{2}.

**2. Point of intersection of tangents at any two points on the parabola: **The point of intersection of tangents drawn at two different points *P(at*_{1}^{2}, 2*at*_{1}) and *Q(at*_{2}^{2}, 2*at*_{2}) on the parabola *y*^{2} = 4*ax * is *(at*_{1}*t*_{2}, a(*t*_{1}+*t*_{2})).

**3. Angle of intersection of tangents at any two points on the parabola:** The angle between the tangents drawn at two different points *P*(*at*_{1}^{2}, 2*at*_{1}) and *Q*(*at*_{2}^{2}, 2*at*^{2}) on the parabola *y*^{2} = 4*ax* is given by

**4**. **Properties of Tangents to a Parabola:**

- If P is any point on a parabola y
^{2}= 4ax such that the tangent at P meets the axis of the parabola at T, then OT = OM where PM is the ordinate at P.

- In the above figure, we have:

* FP = FT*

Where F is the focus of the parabola.

- Perpendicular tangents of a parabola intersect at a point on the directrix of the parabola.
- The tangents at the ends of a focal chord of a parabola intersect at right angles and so the point of intersection of these tangents lies on the directrix.
- The circumcircle of the triangle formed by any three tangents to a parabola passes through the focus of the parabola.
- The orthocenter of the triangle formed by any three tangents to a parabola lies on the directrix of the parabola.

**5. Equation of the pair of tangents from a point a parabola:** The combined equation of the pair of tangents drawn from a point *P*(*x*_{1},*y*_{1}) to the parabola *S **º** y*^{2} – 4*ax* = 0 is

Where* S = 0 *is the equation of the parabola i.e. *S **º** ** y*^{2} – 4*ax*;

*S*’ is obtained by replacing *x* by *x*_{1} and y by *y*_{1} in *S* i.e.

*S’ **º** ** y*_{1}^{2} – 4*ax*_{1} and

*T **º** ** yy*_{1} – 2*a* (*x* + *x*_{1}) is obtained by replacing *x*^{2} by *xx*_{1}, *y*_{2} by *yy*_{1 }, *x* by in *S*.

**6. Equation of the chord of contact of tangents to a parabola:** The equation of the chord of contact *QR* of tangents *PQ* and *PR* drawn from a point *P*(*x*_{1}, *y*_{1}) to the parabola *y*^{2} = 4*ax* is given by

**Note:** The equation of the chord of contact of tangents from a point *P*(*x*_{1},*y*_{1}) outside the parabola is the same as the equation of the tangent at a point *P*(*x*_{1},*y*_{1}) on the parabola.

Also Read : Atomic Models in Chemistry

(For more updated content related to current affairs, history, polity, geography, economics, mathematics and general sciences for various competitive examinations, follow us on www.rsaggarwal.com . You can also buy preparatory material and books for these examinations at our website, www.radianbooks.in.)