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.
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