Question

Form the differential equation that represents all parabolas each of which has a latus rectum 4a and whose axes are parallel to the x-axis

Answer 1

Equation of parabola whose axis is parallel to the x-axis with latus rectum 4a is

(y – β)² = 4a(x – α)  ........(1)

Here (α, β) is the vertex of the parabola.

Differentiating (1) w.r.t x, we get

`2(y - beta) ("d"y)/("d"x)` = 4a  .........(2)

Again, differentiating (2) w.r.t x, we get

`2[(y - beta) ("d"^2y)/("d"x^2) + (("d"y)/("d"x))^2]` = 0  ........(3)

From (2) we have,

`(y - beta) ("d"y)/("d"x)` = 2a

`y - beta = (2"a")/(("d"y)/("d"x))`

Using this in (3) we get

`(2"a")/(("d"y)/("d"x)) ("d"^2y)/("d"x^2) + (("d"y)/("d"x))^2` = 0

or

`2"a" ("d"^2y)/("d"x^2) + (("d"y)/("d"x))^3`

= 0 is the required differential equation