TS EAMCET · Maths · Straight Lines

If the tangent at a point $P$ on the curve $y = 4 x^{4} + x$ is perpendicular to the tangent to the same curve at $(0, 0),$ then the point $P$ is

A $(\frac{- 1}{2}, \frac{- 1}{4})$
B $(\frac{1}{2}, \frac{3}{4})$
C $(1, 5)$
D $(- 1, 3)$

Verified Solution

Answer & Solution

Correct Answer

(A) $(\frac{- 1}{2}, \frac{- 1}{4})$

Step-by-step Solution

Detailed explanation

Given y=4x4+x Differentiating the above equation, ⇒dydx=16x3+1 The tangent at point (0, 0) has a slope =160+1=1 The tangent at point P(a,b) has a slope 16a3+1 m1m2=-1 when the lines with slope m1 and m2 are perpendicular to each other. ∴16a3+11=-1 ⇒a3=-18…

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 the tangent at a point P on the curve y=4x4+x is perpendicular to the tangent to the same curve at (0, 0), then the point P is

Step-by-step Solution

If the tangent at a point $P$ on the curve $y = 4 x^{4} + x$ is perpendicular to the tangent to the same curve at $(0, 0),$ then the point $P$ is