Question

At any point \( (x, y) \) on a curve, if the length of the subnormal is \( (x - 1) \) and the curve passes through \( (1, 2) \), then the curve is a conic. A vertex of the curve is:

• A. \( (1, 0) \)
• B. \( (0, 1) \)
• C. \( (\sqrt{5}, 0) \)
• D. \( (0, \sqrt{5}) \)

Hint:

Use geometric definitions like subnormal and derive the differential equation to identify conic types.

Answer 1

The Correct Option is D

The length of the subnormal is given by:
\[ \text{subnormal} = y \cdot \frac{dy}{dx} = x - 1 \] Thus, the differential equation becomes: \[ y \frac{dy}{dx} = x - 1 \Rightarrow y \, dy = (x - 1) \, dx \] Integrate both sides: \[ \frac{y^2}{2} = \frac{(x - 1)^2}{2} + C \Rightarrow y^2 = (x - 1)^2 + C' \] Use point \( (1, 2) \) to find \( C' \): \[ 4 = 0 + C' \Rightarrow C' = 4 \] So the equation is: \[ y^2 = (x - 1)^2 + 4 \Rightarrow (x - 1)^2 - y^2 = -4 \] This is a conic (hyperbola). Put in standard form and find vertex:
Vertex lies at the center, \( (0, \sqrt{5}) \).