If the line a² x + a y + 1 = 0, for some real number a, is normal to the curve x y = 1, then

Correct Answer is : a<0a < 0a<0

Correct Answer is : a<0a < 0a<0

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NIMCET 2024 Paper

Section: Mathematics

Question : 44 of 120

Marks: +1, -0

If the line

a^2 x + a y + 1 = 0

a

x y = 1

Solution:

Let any point on the curve

x y = 1

\left(t, \frac{1}{t}\right)

, to calculate the slope of tangent at this point, let us differentiate the function,

x y' + y = 0 \Rightarrow y' = -\frac{y}{x}

Slope of the tangent

= -\frac{\frac{1}{t}}{t} = -\frac{1}{t^2}

Slope of the normal

= t^2

t^2 > 0

, slope of the normal should be positive.

\Rightarrow -\frac{a^2}{a} > 0 \text{ or } a < 0

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