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.
Answer & explanation
Correct answer: option 2
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.