GATE 2025 MA – Question 39

Calculus · Multivariable Differentiation and Optimization · 2 marks · Multiple choice

C is the intersection z²=x²+y² and 4x+z=7. A point P on C has minimum distance to the xy-plane. Find its distance from the origin.

7/5
$7\sqrt2/5$
14/5
$14\sqrt2/5$

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Correct answer: (B) $7\sqrt2/5$

Explanation

On the cone, |x|≤|z|. Thus $7=|4x+z|\le4|x|+|z|\le5|z|$, so |z|≥7/5. Equality occurs at x=z=7/5,y=0. The origin distance is √(x²+y²+z²)=√2|z|=**7√2/5, B**.

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