Revision: Conic Section Mathematics ISC (Commerce) Class 11 CISCE

A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point (not on the line) in the plane. 

Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola in following fig.

To find the Length of the latus rectum of the parabola `y^2` = 4ax in following fig.

By the definition of the parabola, AF = AC.
But AC = FM = 2a
Hence AF = 2a.
And since the parabola is symmetric with respect to x-axis AF = FB and so
AB = Length of the latus rectum = 4a.

An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant.

Latus rectum of an ellipse is a line segment perpendicular to the major axis through any of the foci and whose end points lie on the ellipse in following fig.

To find the length of the latus rectum of the ellipse `x^2/a^2 + y^2/b^2 = 1`
Let the length of `AF_2` be l.
Then the coordinates of A are (c, l ),i.e., (ae, l )
Since A lies on the ellipse  `x^2/a^2 + y^2/b^2 = 1`,
`(ae)^2/a^2+l^2/b^2=1`

`=> l^2 = b^2(1-e^2)`

But `e^2 = c^2/a^2 = (a^2 - b^2)/a^2 = 1- b^2/a^2`

Therefore `l^2 = b^4/a^2, i.e., l = b^2/a`
Since the ellipse is symmetric with respect to y-axis ,`AF_2` = `F_2B` and so length of the latus rectum is `(2b)^2/a.`

A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant.