Question

Prove that curves xy=a² and x²+y²=2a² touch each-other.

Answer 1

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.