First slide

Tangents and normals

Question

Sum of the intercepts on the co-ordinate axes of any tangent to $\sqrt{x} + \sqrt{y} = \sqrt{a}$ is:

Easy

A

$a$
B

$\sqrt{a}$
C

$a^{2}$
D

None

Solution

$\sqrt{x} + \sqrt{y} = \sqrt{a} \Rightarrow \frac{d y}{d x} = - \frac{\sqrt{y}}{\sqrt{x}}$
$∴$ Equation of tangent at $(x_{1}, y_{1})$ is $y - y_{1} = - \frac{\sqrt{y_{1}}}{\sqrt{x_{1}}} (x - x_{1})$

$α =$ intercept on $x -$ axis $= \sqrt{a} \sqrt{x_{1}}$

$β =$ intercept on $y -$ axis $= \sqrt{a} \sqrt{y_{1}}$

$∴$ $α + β = \sqrt{a} (\sqrt{x_{1}} + \sqrt{y_{1}}) = \sqrt{a} \sqrt{a} = a$ .

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