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Equation of Tangent

Equation of tangent at point P(x1, y1) to parabola y2 = 4ax
yy1 – 2a(x +x1) = 0 or T = 0  ...(5)
where T is an expression which we get by replacing y2 by yy1 and 2x by x + x1.
Equation of tangent at point P(t) or P(at2, 2at) In (5) replacing y1 by 2at and x1 by at2, we have equation of tangent,
2at y = 2a(x + at2) or ty = x + at2  ....(6)
Here slope of tangent m = 1/t.
Equation of tangent in slope (m) form In (6) replacing t by 1/m we have y = mx + 70851.png which is equation of tangent in terms of slope and the point of contact is 70845.png.
Thus if line y = mx + c touches parabola y2 = 4ax we must have c = a/m (comparing equation with y = mx + a/m).
Equation of parabolas
Point of contact in terms of slope (m)
Equation of tangent in terms of slope (m)
Condition of tangency
y2 = 4ax
y = mx + 70815.png
c = 70809.png
y2 = –4ax
y = mx70797.png
c = –70791.png
x2 = 4ay
y = mx – am2
c = –am2
x2 = –4ay
(–2am, –am2)
y = mx + am2
c = am2

Properties of tangents

  1. Point of intersection of tangents at two points P(t1) and Q(t2) on the parabola is (at1t2, a (t1 + t2)).
  2. Locus of foot of perpendicular from focus upon any tangent is tangent at vertex.
  3. Length of tangent between the point of contact P(t) and point where it meets the directrix Q subtends right angle at focus.
  4. Tangents at extremities of focal chord are perpendicular and intersect on directrix.
Pair of tangents from point (x1, y1) lying outside parabola Pair of tangents from external point (x1, y1) is given by (y2 – 4ax) 70778.png(yy1 – 2a (x + x1)}2 or SS1 = T2, where, S = y2 – 4ax, S1 = y12 – 4ax1 and T = yy1 – 2a (x + x1).

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