Question

The order of the differential equation of all parabolas, whose latus rectum is 4a and axis parallel to the x axis is

• A. one
• B. four
• C. three
• D. two

Answer 1

Answer: (D)

two

Answer 2

The general equation of a parabola whose axis is parallel to the x-axis is given by:

$(y - k)^2 = 4A(x - h)$

where $(h, k)$ is the vertex of the parabola and $4A$ is the length of the latus rectum.

According to the problem statement, the latus rectum of the parabolas is $4a$. This means $4A = 4a$, where 'a' is a given constant (a fixed length). Therefore, the equation of the family of parabolas becomes:

$(y - k)^2 = 4a(x - h)$

In this equation, $h$ and $k$ are arbitrary constants because they can vary for different parabolas within this family (different vertices). The value 'a' is a fixed parameter defining the latus rectum length, not an arbitrary constant that varies from one member of the family to another.

The order of a differential equation is equal to the number of independent arbitrary constants in the general solution (or the equation of the family of curves). In the given equation $(y - k)^2 = 4a(x - h)$, there are two independent arbitrary constants: $h$ and $k$.

Therefore, the order of the differential equation for this family of parabolas will be 2.