Equation of Tangent
Equation of tangent at point P(x_{1}, y_{1}) to parabola y^{2} = 4ax
yy_{1} â€“ 2a(x +x_{1}) = 0 or T = 0 ...(5)
where T is an expression which we get by replacing y^{2} by yy_{1} and 2x by x + x_{1.}
Equation of tangent at point P(t) or P(at^{2}, 2at) In (5) replacing y_{1} by 2at and x_{1} by at^{2}, we have equation of tangent,
2at y = 2a(x + at^{2}) or ty = x + at^{2 }....(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 + which is equation of tangent in terms of slope and the point of contact is .
Thus if line y = mx + c touches parabola y^{2} = 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

y^{2} = 4ax


y = mx +

c =

y^{2} = â€“4ax


y = mx â€“

c = â€“

x^{2} = 4ay

(2am, am^{2})

y = mx â€“ am^{2}

c = â€“am^{2}

x^{2} = â€“4ay

(â€“2am, â€“am^{2})

y = mx + am^{2}

c = am^{2}

Properties of tangents
 Point of intersection of tangents at two points P(t_{1}) and Q(t_{2}) on the parabola is (at_{1}t_{2}, a (t_{1} + t_{2})).
 Locus of foot of perpendicular from focus upon any tangent is tangent at vertex.
 Length of tangent between the point of contact P(t) and point where it meets the directrix Q subtends right angle at focus.
 Tangents at extremities of focal chord are perpendicular and intersect on directrix.
Pair of tangents from point (x_{1}, y_{1}) lying outside parabola Pair of tangents from external point (x_{1}, y_{1}) is given by (y^{2} â€“ 4ax) (yy_{1} â€“ 2a (x + x_{1})}^{2} or SS_{1} = T^{2},^{ }where, S = y^{2} â€“ 4ax, S_{1} = y_{1}^{2} â€“ 4ax_{1} and T = yy_{1} â€“ 2a (x + x_{1}).