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GATE 2022 AE – Question 11

Engineering Mathematics · Calculus: Limits, continuity, differentiability, chain rule, maxima and minima, integration · 1 mark · Multiple choice

The tangent to $y=x^2$ at (1,1) is

  1. $y=2x-2$
  2. $x=2y-1$
  3. $y-1=2(x-1)$
  4. $x-1=2(y-1)$

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Correct answer: (C) $y-1=2(x-1)$

Explanation

**Slope of the tangent.** For $y=x^2$, the derivative is $\dfrac{dy}{dx}=2x$. At $x=1$ the slope is $m=2$.

**Equation of the tangent** through the point $(1,1)$ with slope 2 (point-slope form):
$$y-1=2(x-1)\;\Rightarrow\;y=2x-1 .$$

Option C, $y-1=2(x-1)$, is exactly this line. (Option A, $y=2x-2$, has the right slope but does not pass through $(1,1)$.)