Prove that curves xy=a² and x²+y²=2a² touch each-other.
Answers
we have to prove that curves xy = a² and x² + y² = 2a² touch each other.
concept : two curves each other, it means slope of tangent of curves must be equal at the common point.
so, first find common point.
solve equation xy = a² and x² + y² = 2a²
x² + (a²/x)² = 2a²
⇒x² + a⁴/x² = 2a²
⇒x⁴ + a⁴ = 2a²x²
⇒(x² - a²)² = 0
⇒x = ±a
hence, y = a²/x = ±a
so, there are two common points. these are ; (a, a) and (-a, -a).
now find slope of tangent of both curves at these points.
xy = a²
differentiate with respect to x,
x dy/dx + y = 0
⇒dy/dx = -y/x
slope of tangent of curve xy = a², dy/dx = -y/x
putting common points we get, dy/dx = -(a/a) = -(-a)/(-a) = -1 ....(1)
similarly, differentiate x² + y² = 2a² with respect to x,
2x + 2y dy/dx = 0
dy/dx = -x/y
so, slope of tangent of curve x² + y² = 2a², dy/dx = -x/y
putting common points we get, dy/dx = -(a)/(a) = -(-a)/(-a) = -1 .....(2)
from equations (1) and (2), it is clear that slope of tangent of the curves at the common points is equal.
hence, curves xy = a² and x² + y² = 2a² touch each other.