Question

find the radius of curvature of a rectangular hyperbola xy= 12 at (3,4)​

Answer 1

Answer:

Radius of curvature = 125/24

Step-by-step explanation:

The given curve is

xy = 12 ..... (1)

Differentiating both sides with respect to x, we get

x dy/dx + y = 0 ..... (2)

Again differentiating both sides with respect to x, we get

x d²y/dx² + dy/dx + dy/dx = 0

or, x d²y/dx² + 2 dy/dx = 0

or, d²y/dx² = - 2/x * dy/dx ..... (3)

From (2) and (3), using (1), we get

dy/dx = - y/x

d²y/dx² = 2y/x²

Therefore radius of curvature is

= {1 + (dy/dx)²}^(3/2) / (d²y/dx²)

= (1 + y²/x²)^(3/2) / (2y/x²)

= {(x² + y²)^(3/2) / x³} * (x² / 2y)

= (x² + y²)^(3/2) / (2xy)

= (3² + 4²)^(3/2) / (2 * 3 * 4) at (3, 4)

= 125 / 24 at (3, 4)

Answer 2

Answer:Radius of curvature = 125/24

Step-by-step explanation:

The given curve is

xy = 12 (1)

Differentiating both sides with respect to x, we get

x dy/dx + y = 0 . (2)

Again differentiating both sides with respect to x, we get

x d²y/dx² + dy/dx + dy/dx = 0

or, x d²y/dx² + 2 dy/dx = 0

or, d²y/dx² = - 2/x * dy/dx (3)

From (2) and (3), using (1), we get

dy/dx = - y/x

d²y/dx² = 2y/x²

Therefore radius of curvature is

= {1 + (dy/dx)²}^(3/2) / (d²y/dx²)

= (1 + y²/x²)^(3/2) / (2y/x²)

= {(x² + y²)^(3/2) / x³} * (x² / 2y)

= (x² + y²)^(3/2) / (2xy)

= (3² + 4²)^(3/2) / (2 * 3 * 4) at (3, 4)

= 125 / 24 at (3, 4)

Step-by-step explanation: