Question

The curve passing through the point (1, 2) given that the slope of the tangent at any point (x,y) is 2x/y represents​ which geometric figure??

Answer 1

The curve passing through the point (1, 2) given that slope of the tangent at any point (x, y) is 2x/y.

To find : The equation of curve ( or representation of geometric figure )

Solution : let y = f(x) is the curve then dy/dx is the slope of tangent of curve at the point (x, y).

so dy/dx = 2x/y

⇒ydy = 2x dx

⇒∫y dy = 2∫ x dx

⇒y²/2 = x² + k

curve passing through the point (1,2)

so, (2)²/2 = (1)² + k ⇒k = 1

so the curve is y²/2 = x² + 1 ⇒y²/(√2)² - x²/1² = 1

Therefore the required of curve is y²/(√2)² - x²/1² = 1 , a hyperbolic equation.