Tangent to a Parabola

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

(i) Point Form: The equation of the tangent to a parabola y2 = 4ax at a point P(x1, y1) on the parabola is

yy1 = 2a (x+x1)  

Note: The equation of the tangent to a parabola (bx – ay)2 + 2gx + 2fy + d = 0 at a point P(x1, y1) is obtained by replacing:

(ii) Slope Form: The equation of line which has a slope m and is tangent to the parabola y2 = 4ax is

(ii) Condition of tangency:

(b) A line lx + my + n = 0 is tangent to parabola y2 = 4ax if ln = am2.

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(at12, 2at1) and Q(at22, 2at2) on the parabola y2 = 4ax  is (at1t2, a(t1+t2)).

3. Angle of intersection of tangents at any two points on the parabola: The angle between the tangents drawn at two different points P(at12, 2at1) and Q(at22, 2at2) on the parabola y2 = 4ax is given by

4. Properties of Tangents to a Parabola:

  • If P is any point on a parabola y2 = 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(x1,y1) to the parabola S º y2 – 4ax = 0 is    

Where S = 0 is the equation of the parabola i.e. S º  y2 – 4ax;

S’ is obtained by replacing x by x1 and y by y1 in S i.e.

S’ º  y12 – 4ax1 and

T º  yy1 – 2a (x + x1) is obtained by replacing x2 by xx1, y2 by yy1 , 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(x1, y1) to the parabola y2 = 4ax is given by

Note: The equation of the chord of contact of tangents from a point P(x1,y1) outside the parabola is the same as the equation of the tangent at a point P(x1,y1) on the parabola.

Also Read : Atomic Models in Chemistry

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