Question

The order of the differential equation representing the family of all parabolas having a fixed latus rectum of length \(4a\) and an axis parallel to the x-axis is...

• A. \(1\)
• B. \(2\)
• C. \(3\)
• D. \(4\)

Hint:

The order equals the number of independent arbitrary constants in the family.

Answer 1

The Correct Option is B

Step 1: Differentiate once
From $(y-k)^{2}=4a(x-h)$: $2(y-k)y'=4a$.

Step 2: Differentiate again
$2y'^{2}+2(y-k)y''=0$, which still contains $k$.

Step 3: Eliminate k
From the first step $y-k=\frac{2a}{y'}$. Substituting gives $y'^{3}+2ay''=0$, free of $h$ and $k$ and containing $y''$.

Final Answer:
So the order is 2. \[ \boxed{\text{(B) }2} \]