Question

If the lengths of the tangent, subtangent, normal, and subnormal for the curve \( y = x^2 + x - 1 \) at the point \( (1,1) \) are \( a, b, c, \) and \( d \) respectively, then their increasing order is:

• A. \( b, a, c, d \)
• B. \( b, a, c, d \)
• C. \( a, b, c, d \)
• D. \( b, a, d, c \)

Hint:

For problems involving tangent and normal lengths, use the standard formulas: \[ a = \frac{1 + m^2}{m}, b = \frac{x}{m}, c = \frac{1 + m^2}{1}, d = \frac{x}{1} \] Evaluating at the given point simplifies ranking.

Answer 1

The Correct Option is D

Differentiating \( y = x^2 + x - 1 \): \[ \frac{dy}{dx} = 2x + 1 \] Evaluating at \( x = 1 \): \[ m = 2(1) + 1 = 3 \] Using standard formulas: - Length of tangent \( a = \frac{1 + m^2}{m} \) - Length of subtangent \( b = \frac{x}{m} \) - Length of normal \( c = \frac{1 + m^2}{1} \) - Length of subnormal \( d = \frac{x}{1} \) After substitution and solving: \[ b<a<d<c \] Thus, the correct order is: \[ b, a, d, c \]