Two statements are given, one labelled Assertion (A) and the other labelled Reason (R).

Suppose $y^2 = ax^3 + b$ is a curve such that slope of tangent to the curve at point (2, 3) is 4.

Assertion (A): The equation of tangent to the curve $y^2 = ax^3 + b$ at (2, 3) is $y = 4x - 5$.
Reason (R): $a = 2$ and $b = -7$.

Select the correct answer from the options given below.

Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).

The correct answer is Option (2) → Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).

$y^2 = ax^3+b$

∵ (2, 3) lies on the curve

$⇒3^2 = a × 2^3+ b⇒ 8a+ b = 9$   ...(i)

Also, slope of tangent to the curve at (2, 3) is 4

$⇒2y\frac{dy}{dx}=3ax^2⇒\frac{dy}{dx}=\frac{3ax^2}{2y}$

Now, $\left(\frac{dy}{dx}\right)_{(2,3}= 4⇒\frac{3a×2^2}{2×3}= 4⇒ a = 2$.

Putting $a = 2$ in equation (i), we get

$8×2+b=9⇒b=-7$

∴ Reason is true.

Now, equation of tangent to the curve at (2, 3) is

$(y-3)=4(x-2)⇒y=4x-5$

∴ Assertion is true.

Hence, both Assertion and Reason are true but Reason is not the correct explanation of Assertion.