If a variable tangent to the curve makes intercepts a, b on x and y axis respectively, then the value of is

JEE MainMathematicsToughApplication of derivatives

If a variable tangent to the curve $x^{2}y = c^{3}$ makes
intercepts a, b on x and y axis respectively, then the value of $a^{2}b$
is

Show answer and solution

x^{2}y = c^{3}

\Rightarrow y = \frac{c^{3}}{x^{2}}

\Rightarrow \frac{dy}{dx} = - \frac{2c^{3}}{x^{3}}

Variable tangent:

(y - y_{1}) = - \frac{2c^{3}}{x_{1}^{3}}(x - x_{1})

a = \frac{y_{1}x_{1}^{3}}{2c^{3}} + x_{1}

= \frac{x_{1}}{2} + x_{1}

= \frac{3x_{1}}{2}

b = y_{1} + \frac{2c^{3}}{x_{1}^{2}} = \frac{3c^{3}}{x_{1}^{2}}

\Rightarrow a^{2}b = \frac{9}{4}x_{1}^{2} \times \frac{3c^{3}}{x_{1}^{2}} = \frac{27}{4}c^{3}

Source: Grass-G / M-20 Mathematics Paper 12 Dec.docx

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