### Definition and terminology

A *parabola* is the locus of a point, whose distance from a fixed point (focus) is equal to perpendicular distance from a fixed straight line (directrix).

**Focal Distance**: The distance of a point on the parabola from the focus.

**Focal Chord**: A chord of the parabola, which passes through the focus.

**Latus Rectum**: A double ordinate passing through the focus or a focal chord perpendicular to the axis of parabola is called the *Latus Rectum* (L.R.).

- The general equation of a conic is ax
^{2}+ 2hxy + by^{2}+ 2gx + 2fy +c =0. Here if e = 1 and D≠ 0, then it represents a - The general equation of parabola is (y-y
_{0})^{2}= (x-x_{0}), which has its vertex at (x_{0}, y_{0}). - The general equation of parabola with vertex at (0, 0) is given by y
^{2}= 4ax, and it opens rightwards. - The parabola x
^{2}= 4ay opens upwards. - The equation y
^{2}= 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 (at^{2}, 2at) and (at^{2},-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 = Ax
^{2}+ 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 = Ay
^{2}+ 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 (x
_{1}, y_{1}) lies outside, on or inside the parabola y^{2}= 4ax, according as the expression y_{1}^{2}= 4ax_{1}is positive, zero or negative. - Length of the chord intercepted by the parabola on the line y = mx + c is (4/m
^{2})√a(1 + m^{2}) (a – mc). - Length of the focal chord which makes an angle δ with the x-axis is 4a cosec
^{2}δ. - In parametric form, the parabola is represented by the equations x = at
^{2}and y = 2at. - The equation of a chord joining t
_{1}and t_{2}is given by 2x – (t_{1}+ t_{2}) y + 2at_{1}t_{2}= 0 - If a chord joining t
_{1}, t_{2}and t_{3}, t_{4}pass through a point (c, 0) on the axis, then t_{1}t_{2}= t_{3}t_{4}= -c/a - The length of the focal chord having parameters t
_{1}and t_{2}for its end points is a(t_{2 }– t_{1})^{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 y**^{2}= 4ax**Point Form:**

yy_{1} = 2a(x+x_{1}) at the point (x_{1}, y_{1})

**Slope Form:**

y = mx + a/m ( m ≠ 0) at (a/m^{2}, 2a/m)

**Parametric Form:**

ty = x + at^{2} at point (at^{2}, 2at)

- The coordinates of points of intersection of the tangents at two points P(at
_{1}^{2}, 2at_{1}) and Q(at_{2}^{2}, 2at_{2}) are S(at_{1}t_{2}, a(t_{1}+ t_{2})). - ?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 y**^{2}= 4ax**Point form**

y-y_{1} = -y_{1}/2a(x-x_{1}) at the point (x_{1}, y_{1})

**Slope Form**

y = mx – 2am – am^{3} at (am^{2}, -2am)

**Parametric Form**

y + tx = 2at + at^{3} at (at^{2}, 2at)

- The point of intersection of normals at any two points say P(at
_{1}^{2}, 2at_{1}) and Q(at_{2}^{2}, 2at_{2}) on the parabola is given by R(2a + a(t_{1}^{2}+ t_{2}^{2}+ t_{1}t_{2}), -at_{1}t_{2}(t_{1 }+ t_{2})). - 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 y
^{2}= 4ax whose middle point is P(x_{1},y_{1}) is

yy_{1} – 2a(x – x_{1}) = y_{1}^{2} – 4ax_{1}.

- The equation of pair of tangents from the point P(x
_{1}, y_{1}) to the parabola y^{2}= 4ax is given by

[yy_{1} – 2a(x + x_{1})]^{2} = (y^{2} – 4ax)(y_{1}^{2} – 4ax_{1}).

- The equation of the polar of the point P(x
_{1},y_{1}) to the parabola y^{2}= 4ax is y_{1}= 2a(x + x_{1}). - The pole of the line lx + my + n = 0 to the parabola y
^{2}= 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 y
^{2}= 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 y
^{2}= 4ax lies on then directrix and has the coordinates (–a, a(t_{1}+ t_{2 }+ t_{3}+ t_{1}t_{2}t_{3})). - 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(x
^{2}+ y^{2}) – 2(h + 2a)x – ky = 0.

## Parabola

**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:

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