The number of values of c such that the straight line $3 x + 4 y = c$ touches the curve $\frac{x^{4}}{2} = x + y$ , is :

0

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1

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2

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4

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Solution

The correct option is A $1$
Equation of curve is $x^{4} = 2 x + 2 y$
$\frac{d y}{d x} = 2 x^{3} - 1$
Equation of line is $3 x + 4 y = c$
Slope of line is $- \frac{3}{4}$
Since, the line touches the curve,
Slope of curve $=$ Slope of tangent
$2 x^{3} - 1 = - \frac{3}{4}$
$\Rightarrow x = \frac{1}{2}$
$\Rightarrow y = - \frac{15}{32}$
$∴$ only one value of c is possible

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3 x + 4 y = c

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