Tangents to a Parabola and Their Properties
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Direct answer
Every tangent to the parabola y^2 = 4ax can be written in three interchangeable ways: at a point (x1, y1) on the curve it is yy1 = 2a(x + x1); at the parametric point (at^2, 2at) it is ty = x + at^2; and in slope form it is y = mx + a/m with m ≠ 0, touching at (a/m^2, 2a/m). Two tangents at parameters t1 and t2 intersect at (a t1 t2, a(t1 + t2)). For a focal chord t1 t2 = -1, so the tangents at its ends meet on the directrix x = -a with slopes multiplying to -1 — they are perpendicular. The foot of the perpendicular from the focus onto any tangent lies on the tangent at the vertex, which encodes the reflection property that makes parabolas focus parallel rays.
What you must remember
- Three tangent forms: yy1 = 2a(x + x1) at (x1, y1); ty = x + at^2 at t; y = mx + a/m in slope form, with contact point (a/m^2, 2a/m).
- No horizontal tangent: m = 0 is impossible in the slope form; the tangent at the vertex is the vertical line x = 0.
- Intersection of tangents at t1 and t2: the point (a t1 t2, a(t1 + t2)); the chord joining the contact points is the focal chord precisely when t1 t2 = -1.
- Perpendicular tangents meet on the directrix: slopes 1/t1 and 1/t2 multiply to 1/(t1 t2) = -1 for focal chords — the directrix is the parabola's degenerate director circle.
- Latus rectum ends (a, ±2a): tangents there are y = x + a and y = -x - a, crossing at (-a, 0) on the directrix at right angles — a one-line verification worth memorising.
- Focus-foot property: the perpendicular from focus (a, 0) onto any tangent lands on x = 0, the tangent at the vertex.
- Reflection property: the normal at a point bisects the angle between the focal distance and the axis-parallel through that point; rays parallel to the axis reflect through the focus.
Chasing an intersection
Derive the meeting point of the tangents at t1 and 2 for y^2 = 4ax. The tangents are y = x + a (from ty = x + at^2 with t = 1) and 2y = x + 4a, that is, y = x/2 + 2a. Setting them equal: x + a = x/2 + 2a, so x = 2a and y = 3a — matching the formula (a t1 t2, a(t1 + t2)) = (2a, 3a). Now push the logic to the focal result: for t1 t2 = -1 the x-coordinate is -a, fixed on the directrix, while the product of the slopes 1/t1 and 1/t2 equals -1, forcing perpendicularity. So the locus of intersections of perpendicular tangents is exactly the directrix — asked repeatedly as a locus question, and answerable in one line once this derivation is internalised. Notice also what the intersection point does geometrically: it is the pole of the chord joining t1 and t2, which is why chord-of-contact and tangent questions share one machinery.
Where students slip
JEE Main samples this chapter with slope-form plug-ins (find the tangent with slope 2, or the tangent parallel to a given line) and locus asks like "locus of intersection of perpendicular tangents" — answer x = -a. Advanced layers the focal-chord perpendicularity with parametric ranges or asks for common tangents to a parabola and a circle. Three slips dominate. First, sign errors in the slope form: y^2 = 4ax takes y = mx + a/m, while y^2 = -4ax takes y = mx - a/m — mixing orientations costs the whole question. Second, writing yy1 = 2a(x + x1) for a point not on the parabola; the line is then the polar (chord of contact) of that point, not a tangent, a distinction Advanced explicitly exploits. Third, forgetting m = 0 is banned and m infinite corresponds to x = 0. The topic is core conic-section syllabus for both exams.
Frequently asked questions
What is the tangent to y^2 = 4ax at the point (at^2, 2at)?
ty = x + at^2, with slope 1/t; this parametric form is the fastest route in most problems.
How is the tangent written purely in terms of slope?
y = mx + a/m with m ≠ 0, touching the parabola at (a/m^2, 2a/m).
Where do two perpendicular tangents to a parabola meet?
Always on the directrix x = -a, the degenerate director circle of the parabola.
What is special about tangents at the ends of a focal chord?
They intersect at right angles on the directrix, because t1 t2 = -1 makes the product of their slopes -1.
Is yy1 = 2a(x + x1) a tangent for every point (x1, y1)?
Only when (x1, y1) lies on the parabola; otherwise the line is the polar of that point, the chord of contact of the two tangents drawn from it.