If the equation of the tangent at (2, 3) on y2 = ax3 + b is y = 4x - 5, then the value of a2 + b2 is:
If the equation of the tangent at (2, 3) on y2 = ax3 + b is y = 4x - 5, then the value of a2 + b2 is:
Show Hint
- \(51\)
- \(53\)
- \(58\)
- \(25\)
The Correct Option is B
Solution and Explanation
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.
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