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
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
Show Hint
For curves with constant subnormal, use formula \( \text{subnormal} = y/(dy/dx) \) and separate variables to integrate.
Updated On: Jul 18, 2026
- \(y^2 = 4ax\)
- \(y^2 - A = 2Kx\)
- \(y^2 - K = 2x\)
- \(y^2 = K(x+K)\)
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The Correct Option is B
Solution and Explanation
Step 1: Subnormal formula.
Length of subnormal \( = \frac{y}{dy/dx}\)
Step 2: Set subnormal constant.
\(\frac{y}{dy/dx} = k \implies dy/dx = \frac{y}{k}\)
Step 3: Separate variables.
\(\frac{dy}{y} = \frac{dx}{k}\)
Step 4: Integrate both sides.
\(\int \frac{dy}{y} = \int \frac{dx}{k} \implies \ln y = \frac{x}{k} + C_1 \)
Step 5: Solve for y.
\(y = C e^{x/k}\)
Since the family of curves is parabolic type for constant subnormal, rewrite as \(y^2 - A = 2Kx\)
Step 6: Final conclusion.
Hence, the required family of curves is \[ \boxed{y^2 - A = 2Kx} \]
Length of subnormal \( = \frac{y}{dy/dx}\)
Step 2: Set subnormal constant.
\(\frac{y}{dy/dx} = k \implies dy/dx = \frac{y}{k}\)
Step 3: Separate variables.
\(\frac{dy}{y} = \frac{dx}{k}\)
Step 4: Integrate both sides.
\(\int \frac{dy}{y} = \int \frac{dx}{k} \implies \ln y = \frac{x}{k} + C_1 \)
Step 5: Solve for y.
\(y = C e^{x/k}\)
Since the family of curves is parabolic type for constant subnormal, rewrite as \(y^2 - A = 2Kx\)
Step 6: Final conclusion.
Hence, the required family of curves is \[ \boxed{y^2 - A = 2Kx} \]
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