The slope of tangent at a point `P(x, y)` on a curve is `- x/y`. If the curve passes through the point (3, -4) , find the equation of the curve.

Answer

To solve the problem, we will follow these steps:

1. **Understand the given information**: The slope of the tangent at a point \( P(x, y) \) on the curve is given by the equation \( \frac{dy}{dx} = -\frac{x}{y} \). The curve passes through the point (3, -4).

2. **Separate the variables**: We can rearrange the equation to separate the variables \( y \) and \( x \):
\[ y \, dy = -x \, dx \]

3. **Integrate both sides**: Now we will integrate both sides:
\[ \int y \, dy = \int -x \, dx \]
The left side becomes:
\[ \frac{y^2}{2} \]
and the right side becomes:
\[ -\frac{x^2}{2} + C \]
Therefore, we have:
\[ \frac{y^2}{2} = -\frac{x^2}{2} + C \]

4. **Multiply through by 2 to eliminate the fraction**:
\[ y^2 = -x^2 + 2C \]

5. **Rearrange the equation**: We can rearrange this to:
\[ x^2 + y^2 = 2C \]

6. **Use the initial condition to find \( C \)**: Since the curve passes through the point (3, -4), we can substitute \( x = 3 \) and \( y = -4 \) into the equation:
\[ 3^2 + (-4)^2 = 2C \]
This simplifies to:
\[ 9 + 16 = 2C \implies 25 = 2C \implies C = \frac{25}{2} \]

7. **Substitute \( C \) back into the equation**: Now we substitute \( C \) back into the equation:
\[ x^2 + y^2 = 2 \left(\frac{25}{2}\right) \]
This simplifies to:
\[ x^2 + y^2 = 25 \]

8. **Final equation of the curve**: Thus, the equation of the curve is:
\[ x^2 + y^2 = 25 \]