Definite Integration Question 16
Question: The area of the triangle formed by the tangent to the hyperbola $ xy=a^{2} $ and co-ordinate axes is
[RPET 2000]
Options:
A) $ a^{2} $
B) $ 2a^{2} $
C) $ 3a^{2} $
D) $ 4a^{2} $
Show Answer
Answer:
Correct Answer: B
Solution:
Given $ xy=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={{( \frac{dy}{dx} )} _{(a,a)}}(x-a) $
$ ={{( \frac{-a^{2}}{x^{2}} )} _{(a,a)}}(x-a) $
$ y-a=-(x-a) $ therefore, equation of the tangent at $ (a,a) $ is $ x+y=2a $ .The interception of line $ x+y=2a $ with x-axis is 2a and with y-axis is 2a.
$ \therefore $ Required area = $ \frac{1}{2}\times 2a\times 2a=2a^{2} $ .
BETA
