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Q.

The shortest distance between the curve $y = e x$ and the line y=x is

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a

2 units

b

12 units

c

2units

d

12 units

answer is D.

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Detailed Solution

We know the shortest and greatest distance between two
Curves always lie along the common normal. Let the normal
To the $y = e x at Q α, e α$ meets the line $y = x at P$ .
Question Image
$d y d x = e x ∴ − 1 = − 1 d y d x ⇒ − e α = − 1 ⇒ α = 0 ∴ Q ≡ (0, 1)$
$∴ normal = y − 1 = − 1 (x − 0) ⇒ x + y − 1 = 0 ∴ P Q = 14 + 14 ⇒ 12$

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The shortest distance between the curve y= e x and the line y=x is