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).

• 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(t₂ - 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(t₁ + 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 + t₂ + 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: