Parabola - Practice Questions & MCQ

Parabola

A parabola is the locus of a point moving in a plane such that its distance from a fixed point (focus) is equal to its distance from a fixed line (directrix).

Hence it is a conic section with eccentricity e = 1.

Standard equation of a parabola

Let focus of parabola is S(a, 0) and directrix be x + a = 0

P(x, y) is any point on the parabola.

Now, from the definition of the parabola, 

which is the required equation of a standard parabola

Important Terms related to Parabola

1. Axis: The line which passes through the focus and perpendicular to Directrix of the parabola. For parabola , X-axis is the Axis.

2. Vertex: The point of intersection of the parabola and axis. For parabola , A(0, 0) i.e. origin is the Vertex. 

3. Double Ordinate: Suppose a line perpendicular to the axis of the parabola meets the curve at Q and Q’. Then, QQ’ is called double ordinate of the parabola. 

4. Latus Rectum: The double ordinate LL’ passing through the focus is called latus rectum of the parabola. 

5. Focal Chord: A chord of a parabola which is passing through the focus. In the figure PP’ and LL’ are the focal chord. 

6. Focal Distance: The distance from the focus to any point on the parabola. I.e. PS 

Length of the Latus rectum and Parametric form

The length of the latus rectum of the parabola   is 4a.

L and L’ be the ends of the latus rectum.

Clearly the x-coordinate of L and L' will be 'a'

As L and L' lie on parabola, so put x = a in the equation of parabola to get y coordinates of L and L'

Parametric Equation:

From the equation of the parabola, we can write 

Point P(t) lying on the parabola means the coordinates of P are (at², 2at)

Other Forms of Parabola

1. Parabola with focus S (-a, 0) and directrix x = a (Parabola opening to left)

Equation of the parabola is y² = - 4ax, a > 0

2. Parabola with focus S (0, a) and directrix y = -a (Parabola opening to upward)

Equation of the parabola is x² =  4ay , a > 0.

3. Parabola with focus S (0, -a) and directrix y = a (Parabola opening to downward)

Equation of the parabola is x² =  -4ay , a > 0.

General equation of Parabola

Let S (h, k) be the focus and lx + my + n = 0 be the equation of the directrix and P(x, y) be any point on the parabola.
 

Then, from the definition PS = PM

This is the general equation of a parabola.