Parabola

Definition and terminology

• 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.

 

A parabola is the set of points in a plane that are the same distance from a given point and a given line in that plane. The given point is called the focus, and the line is called the directrix. The midpoint of the perpendicular segment from the focus to the directrix is called the vertex of the parabola. The line that passes through the vertex and focus is called the axis of symmetry (see Figure 1.)

Figure 1. Two possible parabolas.

The equation of a parabola can be written in two basic forms:

• Form 1: y = a( x – h) ² + k
• Form 2: x = a( y – k) ² + h

In Form 1, the parabola opens vertically. (It opens in the “ y” direction.) If a > 0, it opens upward. Refer to Figure 1(a). If a < 0, it opens downward. The distance from the vertex to the focus and from the vertex to the directrix line are the same. This distance is 

A parabola with its vertex at ( h, k), opening vertically, will have the following properties.

• The focus will be at .
• The directrix will have the equation .
• The axis of symmetry will have the equation x = h.
• Its form will be y = a( x – h) ² + k.

In Form 2, the parabola opens horizontally. (It opens in the “ x” direction.) If a > 0, it opens to the right. Refer to Figure 1(b). If a < 0, it opens to the left.

A parabola with its vertex at ( h, k), opening horizontally, will have the following properties.

• The focus will be at .
• The directrix will have the equation .
• The axis of symmetry will have the equation y = k.
• Its form will be x = a( y – k) ² + h.

Example 1

Draw the graph of y = x ². State which direction the parabola opens and determine its vertex, focus, directrix, and axis of symmetry.

The equation y = x ² can be written as

y = 1( x – 0) ² + 0

so a = 1, h = 0, and k = 0. Since a > 0 and the parabola opens vertically, its direction is up (see Figure 2).

Vertex: ( h, k) = (0, 0)

Focus: .

Directrix: .

Axis of symmetry: 

Figure 2. Properties of parabolas.

Example 2

Graph . State which direction the parabola opens and determine its vertex, focus, directrix, and axis of symmetry.

The equation  is the same as .

Since a < 0 and the parabola opens horizontally, this parabola opens to the left (see Figure 3).

Vertex: ( h, k) = (–3, –2)

Focus: 

Directrix: 

Axis of symmetry: 

Figure 3. The graph of Example.

Example 3

Put the equation x = 5 y ² – 30 y + 11 into the form

x = a( y – k) ² + h

Determine the direction of opening, vertex, focus, directrix, and axis of symmetry.

x = 5 y ² – 30 y + 11

Factor out the coefficient of y ² from the terms involving yso that you can complete the square.

x = 5( y ² – 6 y) + 11

Completing the square within the parentheses adds 5(9) = 45 to the right side. Add this amount to the left side to keep the equation balanced.

Subtract 45 from both sides.

x = 5( y – 3) ² – 34

Direction: Opens to the right ( a > 0, opens horizontally)

Vertex: ( h, k) = (–34, 3)

Focus: 

Directrix: 

Axis of symmetry: 

 

1. If the tangent & normal at any point 'PP' of the parabola intersect the axis at TT & GG then ST=SG=SPST=SG=SP where 'SS' is the focus. In other words the tangent and the normal at a point PP on the parabola are the bisectors of the angle between the focal radius SPSP & the perpendicular from PP on the directrix. From this we conclude that all rays emanating from SS will become parallel to the axis of the parabola after reflection.
   
2. The portion of a tangent to a parabola cut off between the directrix & the curve subtends a right angle at the focus, See figure above.
3. The tangents at the extremities of a focal chord intersect at right angles on the directrix, and hence a circle on any focal chord as diameter touches the directrix. Also a circle on any focal radii of a point P(at2,2at)P(at2,2at) as diameter touches the tangent at the vertex and intercepts a chord of length,1+t2−−−−−√1+t2 on normal at the point PP.
   
4. Any tangent to a parabola & the perpendicular on it from the focus meet on the tangent at the vertex.
   
5. lf the tangents at PP and QQ meet in TT, then:
   ⇒TP⇒TP and TQTQ subtend equal angles at the focus SS.
   ⇒ST2=SP.SQ⇒⇒ST2=SP.SQ⇒ The triangles SPTSPT and STQSTQ are similar.
6. Semi latus rectum of the parabola y2−4axy2−4ax, is the harmonic mean between segments of any focal chord of the parabola.
   
7. 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.
8. Length of subtangent at any point P(x, y) on the parabola y2=4axy2=4axequals twice the abscissa of the point P. Note that the subtangent is bisected at the vertex.

9. Length of subnormal is constant for all points on the parabola & is equal to the semi latus rectum. See figure above.

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