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Propositions on Parabola-Part1 Propositions on the Parabola
(i) The tangent at any point P on a parabola bisects the angle between the focal chord through P and the perpendicular from P on the directrix.
The tangent at P (at2, 2at) is ty = x + at2.
It meets the x-axis at T(–at2, 0).
Hence ST = a (1 + t2).
Also, SP = √(a2 (1+t2 )2+4a2 t2 ) = a(1 + t2) = ST, so that
∠MPT = ∠PTS = ∠SPT ⇒ TP bisects ∠SPM.
(ii) The portion of a tangent to a parabola cut off between the directrix and the curve subtends a right angle at the focus.
Let P(at2, 2a), be a point on the parabola y2 = 4ax.
The tangent at P is ty = x + at2.
Point of intersection of the tangent with the directrix x + a = 0 is (–a, at – a/t).
Now, slope of SP is (2at-0)/(at2-a)=2t/(t2-1)
and slope of SK is (at-a/t-0)/(-a-a)=-(t2-1)/2t
⇒ (Slope of the SP).(Slope of SK) = –1.
Hence SP is perpendicular to SK i.e. ∠KSP = 90°.
(iii) Tangents at the extremities of any focal chord intersect at right angles on the directrix.
Let P(at2, 2at) and P(at12, 2at1) be the end points of a focal part on the parabola. Then t.t1 = –1. Equations of the tangents at the point P and the point P’ are ty = x + at2 and t1y = x + at12 respectively.
Let these tangents intersects at a point (h, k). Then h = att1 and k = a(t + t1).
Since the tangents are perpendicular, tt1 = – 1 ⇒ h – a.
Hence the locus of the point (h, k) is x = –a which is the equation of the directrix.
(iv) Any tangent to a parabola and the perpendicular on it from the focus meet on the tangent at the vertex.
Equation of the perpendicular to the tangent ty = x + at2 … (1)
From the focus (a, 0) is tx + y = at. … (2)
and (2) intersect at x = 0 which is the equation of the tangent at the vertex.
Pole and polar of a conic
The locus of the point of intersection of tangents drawn at the extremities of the chord of the conic drawn through a point is called the polar of that point with respect to the conic. This point itself is called the pole. Equation of the polar of a point (x1, y1) with respect to the parabola y2 = 4ax. Let us draw the chord QR from the point P(x1, y1) and if the tangents drawn from point Q and R meet at the point T(h, k), required locus of (h, k) is polar. Since QR is the chord of contact of tangents from (h, k), it’s equation is ky = 2a(x + h) This straight line passes through the point (x1, y1), we have ky1 = 2a(x1 + h) …… (1) Since the relation (1) is true, it follows that point (h, k) always lies on the line. yy1 = 2a(x + x1) …… (2) Hence (2) is the equation to the polar of pole (x1, y1)
Co-normal Points: The three points on the parabola, the normals at which pass through a common point, are called the co-normal points.
Diameter: The locus of the middle point of a system of parallel chords of a parabola is called its diameter.