Question

The area of the triangle formed by the tangent to the hyperbola $x y = a ^ { 2 }$ and co-ordinate axes is

• A. $a ^ { 2 }$
• B. $2 a ^ { 2 }$
• C. $3 a ^ { 2 }$
• D. $4 a ^ { 2 }$

Answer 1

Answer: (B)

$2 a ^ { 2 }$

Answer 2

Given $x y = a ^ { 2 }$ or $y = \frac { a ^ { 2 } } { x }$ .....(i)

There are two points on the curve (a, a),(– a,– a)

The equation of the line at $( a , a )$is,

$y - a = \left( \frac { d y } { d x } \right) _ { ( a , a ) } ( x - a )$ $= \left( \frac { - a ^ { 2 } } { x ^ { 2 } } \right) _ { ( a , a ) } ( x - a )$

$y - a = - ( x - a )$ Therefore, equation of the tangent at $( a , a )$ is $x + y = 2 a$.The interception of line $x + y = 2 a$ with x-axis is 2a and with y-axis is 2a.

$\therefore$ Required area = $\frac { 1 } { 2 } \times 2 a \times 2 a = 2 a ^ { 2 }$.