TS EAMCET · Maths · Application of Derivatives

The ratio of the length of the subnormal to the square of the length of the subtangent at any point $P$ on the curve $y^2=(2 x+1)^3$ is

A 27
B $\frac{1}{9}$
C 9
D $\frac{8}{27}$

Verified Solution

Answer & Solution

Correct Answer

(A) 27

Step-by-step Solution

Detailed explanation

Given equation of curve is $y^2=(2 x+1)^3$ ...(i) Let $P\left(x_1, y_1\right)$ be any point on Eq. (i) So, $ y_1^2=\left(2 x_1+1\right)^3 $ $\Rightarrow \quad\left(2 x_1+1\right)^3 / y_1^2=1$ ...(ii) Now, differentiate Eq. (i) w.r.t. $x$, we get…

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.

Same subject

Match the items given in List-A with those of the items of List-B

	List-A		List-B
(A)	The vertex of the parabola $y^{2} + 4 x - 2 y + 3 = 0$ is	(I)	$(\frac{5}{4}, 1)$
(B)	The vertex of the parabola $x^{2} + 8 x + 12 y + 4 = 0$ is	(II)	$(1, \frac{5}{4})$
(C)	The focus of the parabola $y^{2} - x - 2 y + 2 = 0$ is	(III)	$(\frac{- 1}{2}, 1)$
(D)	The focus of the parabola $x^{2} - 2 x - 8 y - 23 = 0$ is	(IV)	$(1, - 1)$
		(V)	$(- 4, 1)$

The correct match is

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The ratio of the length of the subnormal to the square of the length of the subtangent at any point \(P\) on the curve \(y^2=(2 x+1)^3\) is

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