Question

The equation of the family of curves for which the length of the subnormal at any point \((x,y)\) is always a constant \(k\) is

• A. \(y^2 = 4ax\)
• B. \(y^2 - A = 2Kx\)
• C. \(y^2 - K = 2x\)
• D. \(y^2 = K(x+K)\)

Hint:

For curves with constant subnormal, use formula \( \text{subnormal} = y/(dy/dx) \) and separate variables to integrate.

Answer 1

The Correct Option is B

Step 1: Recall the correct subnormal formula.
For a curve \(y=y(x)\), the length of the subnormal at any point is \(y\cdot\dfrac{dy}{dx}\) (not its reciprocal).

Step 2: Set the subnormal equal to the constant k.
\[ y\frac{dy}{dx}=k \implies y\,dy = k\,dx \]

Step 3: Integrate both sides.
\[ \int y\,dy=\int k\,dx \implies \frac{y^2}{2}=kx+C \]

Step 4: Rearrange into the family form.
\[ y^2=2kx+2C \]
which is exactly of the shape \(y^2-A=2Kx\) with \(A=2C\) and \(K=k\), both arbitrary constants.

Step 5: Final conclusion.
Hence the family of curves is \[ \boxed{y^2-A=2Kx} \]