AP EAMCET · Maths · Application of Derivatives

If $2 y = 3 x - 1$ is a tangent drawn to the curve $y^{2} = a x^{3} + b$ at $(1, 1)$ where $a, b$ are constants, then $(a, b) =$

A $(1, 0)$
B $(0, 1)$
C $(1, - 1)$
D $(- 1, 1)$

Verified Solution

Answer & Solution

Correct Answer

(A) $(1, 0)$

Step-by-step Solution

Detailed explanation

It is given that 2 y=3 x-1 is tangent to y2=ax3+b at a point (1,1) with slope 32 Differentiate the above equation w.r.t. x, 2ydydx=3ax2 dydx=3ax22y At (1,1) dydx=3a(1)22(1)=3a2 The slope is, dydx=32 3a2=32 a=1 The point (1,1) lies on y2=ax3+b 12=1×13+b b=0 The…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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If 2y=3x-1 is a tangent drawn to the curve y2=ax3+b at (1,1) where a, b are constants, then (a, b)=

Step-by-step Solution

If $2 y = 3 x - 1$ is a tangent drawn to the curve $y^{2} = a x^{3} + b$ at $(1, 1)$ where $a, b$ are constants, then $(a, b) =$