Question

The differential equation of all parabolas which have \(4a\) as latus rectum and whose axis is parallel to the X-axis is \(2a\frac{d^{2}y}{dx^{2}}+\left(\frac{dy}{dx}\right)^{k}=0\), then the value of \(k\) is...

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

Hint:

Write the parabola with parameters h and k, differentiate twice, and eliminate h and k.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
A parabola with latus rectum \(4a\) and axis parallel to the X-axis is \((y-k)^{2}=4a(x-h)\), where \(h\) and \(k\) are arbitrary constants.

Step 2: Differentiate once
\(2(y-k)\dfrac{dy}{dx}=4a\), so \(y-k=\dfrac{2a}{y'}\).

Step 3: Differentiate again
\((y-k)y''+(y')^{2}=0\).

Step 4: Eliminate k
Substitute \(y-k=\dfrac{2a}{y'}\): \(\dfrac{2a}{y'}y''+(y')^{2}=0\), i.e. \(2a\,y''+(y')^{3}=0\).

Final Answer:
The power of dy/dx is 3, so k = 3. \[ \boxed{\text{(C) }3} \]