Question

 If the equation of the tangent at (2, 3) on y² = ax³ + b is y = 4x - 5, then the value of a² + b² is:

• A. \(51\)
• B. \(53\)
• C. \(58\)
• D. \(25\)

Hint:

Use the given point on the curve and the slope of the tangent to find the unknown coefficients.

Answer 1

The Correct Option is B

Step 1: Use the point (2, 3) on the curve y² = ax³ + b.
Since (2, 3) lies on the curve y² = ax³ + b, we have: 3² = a(2³) + b
9 = 8a + b ...(1)

Step 2: Differentiate the equation y² = ax³ + b with respect to x.
Differentiating both sides with respect to x, we get: 2y (dy/dx) = 3ax²
dy/dx = (3ax²) / (2y)

Step 3: Find the slope of the tangent at (2, 3).
The slope of the tangent at (2, 3) is given by: (dy/dx) |_{(2, 3)} = (3a(2²)) / (2(3)) = (12a) / 6 = 2a

Step 4: Compare the slope with the given tangent equation.
The given tangent equation is y = 4x - 5.
The slope of this tangent is 4.
Therefore, 2a = 4, so a = 2.

Step 5: Substitute the value of a in equation (1) to find b.
Substitute a = 2 in 9 = 8a + b: 9 = 8(2) + b
9 = 16 + b
b = 9 - 16 = -7

Step 6: Calculate a² + b².
a² + b² = 2² + (-7)² = 4 + 49 = 53

Therefore, the value of a² + b² is 53.