Question

The equation of the curve whose slope is given by \(\frac{dy}{dx}=\frac{4x}{y}\), x>0,y>0 and which passes through the point (2, 2) is: 

• A. \(y^2 = 4x^2-12\)
• B. \(x^2 = 4y^2-12\)
• C. \(y^2 = 4x^2+12\)
• D. \(x^2 = 4y^2+ 12\)

Answer 1

The Correct Option is A

To find the equation of the curve, we start with the given differential equation:

\(\frac{dy}{dx}=\frac{4x}{y}\)

We separate the variables to integrate:

\(y \, dy = 4x \, dx\)

Integrating both sides:

\(\int y \, dy = \int 4x \, dx\)

This results in:

\(\frac{y^2}{2} = 2x^2 + C\)

Solving for \(y^2\), we multiply through by 2:

\(y^2 = 4x^2 + C\)

Now, we use the point (2, 2) to find the constant \(C\):

\(2^2 = 4(2)^2 + C\)

This simplifies to:

\(4 = 16 + C\)

Solve for \(C\):

\(C = 4 - 16 = -12\)

Substitute \(C = -12\) back into the equation:

\(y^2 = 4x^2 - 12\)

Thus, the equation of the curve is:

\(y^2 = 4x^2 - 12\)