The sum of the lengths of the sub-tangent and tangent drawn at the point $(x, y)$ on the curve $y = a log (x^{2} - a^{2})$ varies as

x^{2}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

y^{2}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

x y

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

\frac{y}{x}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B $x y$
Given, $y = a l o g (x^{2} - a^{2})$
$\frac{d y}{d x} = a \cdot \frac{2 x}{x^{2} - a^{2}} = \frac{2 a x}{x^{2} - a^{2}}$
$∴$ Length of subtangent
$= y \frac{d x}{d y}$
$= y \cdot \frac{(x^{2} - a^{2})}{2 a x}$
Length of tangent
$= y \sqrt{1 + {(\frac{d x}{d y})}^{2}}$
$= y \sqrt{1 + {(\frac{x^{2} - a^{2}}{2 a x})}^{2}}$
$= y \sqrt{\frac{4 a^{2} x^{2} + (x^{2} - a^{2})}{4 a^{2} x^{2}}}$
$= y \cdot \frac{y}{2 a x} \sqrt{(x^{2} + a^{2})^{2}} = y \frac{(x^{2} + a^{2})}{2 a x}$
Hence, sum of lengths of tangents & subtangents
$= y \frac{(x^{2} - a^{2})}{2 a x} + y \frac{(x^{2} - a^{2}}{2 a x}$
$= \frac{y}{2 a x} (x^{2} - a^{2} + x^{2} - a^{2})$
$= \frac{x y}{a} = \frac{1}{a} (x y)$
$∴$ Sum of lengths of t & s $α (x y)$ .

Suggest Corrections

Similar questions

The sub-tangent at any point of the curve $x^{m} y^{n} = a^{m + 1}$ varies as

x y = a^{2}

x^{m} . y^{n} = a^{m + n}

y = a log (x^{2} - a^{2})

(a > 0)

(x, y)

y^{2} x^{2} = a^{2} (x^{2} - a^{2})

Join BYJU'S Learning Program