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 parabolax² = 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)

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

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

• 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 parabolay² = 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

• Tangents to the parabola y² = 4ax

1. yy₁ = 2a(x+x₁) at the point (x₁, y₁)

2. y = mx + a/m ( m ≠ 0) at (a/m², 2a/m)

3. ty = x+at² at (at², 2at)

• Normals to the parabola y² = 4ax

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

2. y = mx -2am – am³ at (am², -2am)

3. y + tx = 2at +at³ at (at², 2at)

• 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 orthocenter 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 two vital parabolas along with their basic components like vertex and directrix are tabulated below: