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Revision Notes on Parabola

  • The general equation of a conic is ax2 + 2hxy + by2 + 2gx + 2fy +c =0. Here if e = 1 and D≠ 0, then it represents a parabola.

  • The general equation of parabola is (y-y0)2 = (x-x0), which has its vertex at (x0, y0).

  • The general equation of parabola with vertex at (0, 0) is given by y2 = 4ax, and it opens rightwards. 

  • The parabola x2 = 4ay opens upwards.

  • The equation y2 = 4ax is considered to be the standard equation of the parabola for which the various components are

1. Vertex at (0,0)  

2. Directrix is x + a = 0

3. Axis is y = 0

4. Focus is (a, 0)

5. Length of latus rectum = 4a

6. Ends of latus rectum are L(a, 2a) and L’(a, -2a)

7. The ends of the double ordinate of the parabola can be taken as (at2, 2at) and (at2,-2at).


Parabolay2 = 4ax

  • The parabola y = a(x – h)2 + k has its vertex at (h, k)

  • The perpendicular distance from focus on directrix is half the length of latus rectum

  • For the parabola y = Ax2 + Bx + C, the length of the latus rectum is 1/|A| and axis is parallel to y-axis. If A is positive, then it is concave up parabola and if A is negative then it is concave down parabola.

  • For the parabola x = Ay2 + By + C, the length of the latus rectum is 1/|A| and axis is parallel to x-axis. If A is positive, then it is opening right parabola and if A is negative then it is opening left parabola.

  • Vertex is the middle point of the focus and the point of intersection of directrix and axis.

  • Two parabolas are said to be equal if they have the same latus rectum.

  • The point (x1, y1) lies outside, on or inside the parabola y2 = 4ax, according as the expression y12 = 4ax1 is positive, zero or negative.

  • Length of the chord intercepted by the parabola on the line y = mx + c is (4/m2)√a(1 + m2) (a - mc).

  • Length of the focal chord which makes an angle δ with the x-axis is 4a cosec2δ.

  • In parametric form, the parabola is represented by the equations x = at2 and y = 2at.

  • The equation of a chord joining t1 and t2 is given by 2x – (t1 + t2) y + 2at1t2 = 0

  • If a chord joining t1, t2 and t3, t4 pass through a point (c, 0) on the axis, then t1t2 = t3t4 = -c/a

  • The length of the focal chord having parameters t1 and t2 for its end points is a(t- t1)2.

  • The length of the smallest focal chord of the parabola is 4a, which is the latus rectum of the parabola.

  • Tangents to the parabola y2 = 4ax

  • Point Form:

yy1 = 2a(x+x1) at the point (x1, y1)

  • Slope Form:

y = mx + a/m ( m ≠ 0) at (a/m2, 2a/m)

  • Parametric Form:

ty = x + at2 at point (at2, 2at)

  • The coordinates of points of intersection of the tangents at two points P(at12, 2at1) and Q(at22, 2at2) are S(at1t2, a(t1 + t2)).

  • ?The arithmetic mean of y coordinate of P and Q is the y-coordinate of the point of intersection of tangents at P and Q on the parabola.

  • The geometric mean of the x-coordinate of P and Q is the x-coordinate of the point of intersection of tangents at P and Q on the parabola. 

  • Normal to the parabola y2 = 4ax

  • Point form

y-y1 = -y1/2a(x-x1) at the point (x1, y1)

  • Slope Form

y = mx - 2am – am3 at (am2, -2am)

  • Parametric Form

y + tx = 2at + at3 at (at2, 2at)

  • The point of intersection of normals at any two points say P(at12, 2at1) and Q(at22, 2at2) on the parabola is given by R(2a + a(t12 + t22 + t1t2), -at1t2(t+ t2)).

  • The algebraic sum of the slopes of three concurrent normals is zero. 

  • The algebraic sum of ordinates of the feet of three normals drawn to a parabola form a given point is zero.

  • The centroid of the triangle formed by the feet of three normals lies on the axis of the parabola.

  • The equation of the chord of the parabola y2 = 4ax whose middle point is P(x1,y1) is

yy1 – 2a(x – x1) = y12 – 4ax1.

  • The equation of pair of tangents from the point P(x1, y1) to the parabola y2 = 4ax is given by

[yy1 – 2a(x + x1)]2 = (y2 – 4ax)(y12 – 4ax1).

  • The equation of the polar of the point P(x1,y1) to the parabola y2 = 4ax is y1 = 2a(x + x1).

  • The pole of the line lx + my + n = 0 to the parabola y2 = 4ax is (n/l, -2am/l).

  • The polar of the focus of the parabola is the directrix.

  • Two straight lines are said to be conjugated to each other with respect to a parabola when the pole of one lies on the other. Similarly, two points P and Q are said to be conjugate points if polar of P passes through Q and vice versa.

  • The equation of the diameter of the parabola is y = 2a/m which is parallel to its axis.

  • The length of the tangent to the parabola y2 = 4ax which makes an angle φ with the x - axis is y cosec φ

  • The length of normal of the parabola is y sec φ, where φ is the angle made by the tangent with the x - axis.

  • The length of sub tangent is y cot φ, where φ is the angle made by the tangent with the x - axis.

  • The length of sub normal is y cot φ, where φ is the angle made by the tangent with the x - axis.

  • The foot of the perpendicular from the focus on any tangent to a parabola lies on the tangent at the vertex.

  • The tangents at the extremities of the focal chord intersect at right angles on the directrix.

  • Circle described on the focal length as diameter touches the tangent at the vertex.

  • Circle described on the focal chord as diameter touches the directrix.

  • The equation of the director circle to the parabola is x + a = 0 which is same as the equation of the directrix.

  • The circle circumscribing the triangle formed by any three tangents to a parabola passes through the focus.

  • The orthocentre of any triangle formed by three tangents to a parabola y2 = 4ax lies on then directrix and has the coordinates (–a, a(t1 + t+ t3 + t1t2t3)).

  • The area of the triangle formed by three points on a parabola is twice the area of the triangle formed by the tangents at these points.

  • A circle circumscribing the triangle formed by three co-normal points passes through the vertex of the parabola and its equation is given by 2(x2 + y2) – 2(h + 2a)x - ky = 0.

The vital parabolas along with their basic components like vertex and directrix are tabulated below:

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