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NDA 2020 Maths · Mathematics · Derivatives · medium

If x^2+y^2=r^2 where r is a constant, then (d^2y)/(dx^2 equals

Single correct option

If x2+y2=r2x^2+y^2=r^2 where rr is a constant, then d2ydx2\dfrac{d^2y}{dx^2} equals
  1. A. −1y-\dfrac1y
  2. B. r2y3\dfrac{r^2}{y^3}
  3. C. −xy-\dfrac{x}{y}
  4. D. −r2y3-\dfrac{r^2}{y^3}
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Answer: (D) −r2y3-\dfrac{r^2}{y^3}

Differentiating implicitly, 2x+2yy′=02x+2yy'=0, so y′=−xyy'=-\dfrac xy. Differentiating again, y′′=−y−xy′y2=−y+x2/yy2=−y2+x2y3=−r2y3y''=-\dfrac{y-xy'}{y^2}=-\dfrac{y+x^2/y}{y^2}=-\dfrac{y^2+x^2}{y^3}=-\dfrac{r^2}{y^3}. The option −xy-\dfrac xy is the first derivative, and r2y3\dfrac{r^2}{y^3} has the wrong sign (the circle's upper half is concave, so y′′<0y''<0 for y>0y>0).
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Original question written in the NDA 2020 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake