A line of length a+b moves in such a way that its ends are always on two fixed perpendicular straight lines. Then the locus of point on this line which devides it into two portions of length a and b ,is :

Options

A. A. Parabola
B. B. Circle
C. C. Ellipse
D. D. Hyperbola

Answer

To solve the problem, we need to find the locus of a point on a line of length \( a + b \) that divides it into two segments of lengths \( a \) and \( b \), with the line's endpoints constrained to two fixed perpendicular lines.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let the two fixed perpendicular lines be the x-axis and y-axis.
- The line of length \( a + b \) can be represented with endpoints \( A \) and \( B \) such that the distance \( AB = a + b \).

2. **Positioning the Points**:
- Let point \( A \) be at coordinates \( (x_1, 0) \) on the x-axis and point \( B \) be at coordinates \( (0, y_1) \) on the y-axis.
- The length of the line segment \( AB \) can be expressed using the distance formula:
\[ AB = \sqrt{x_1^2 + y_1^2} = a + b \]

3. **Dividing the Line**:
- Let point \( P \) be the point that divides the line \( AB \) into two segments \( AP = a \) and \( PB = b \).
- By the section formula, the coordinates of point \( P \) can be expressed as:
\[ P\left(\frac{b \cdot x_1}{a + b}, \frac{a \cdot y_1}{a + b}\right) \]

4. **Using the Length Condition**:
- From the distance condition, we have:
\[ x_1^2 + y_1^2 = (a + b)^2 \]

5. **Substituting Coordinates**:
- Substitute \( x_1 = \frac{(a + b) \cdot x}{b} \) and \( y_1 = \frac{(a + b) \cdot y}{a} \) into the distance equation:
\[ \left(\frac{(a + b) \cdot x}{b}\right)^2 + \left(\frac{(a + b) \cdot y}{a}\right)^2 = (a + b)^2 \]

6. **Simplifying the Equation**:
- Simplifying this will yield:
\[ \frac{(a + b)^2 \cdot x^2}{b^2} + \frac{(a + b)^2 \cdot y^2}{a^2} = (a + b)^2 \]
- Dividing through by \( (a + b)^2 \):
\[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \]

7. **Identifying the Locus**:
- The equation \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \) is the standard form of the equation of an ellipse.

### Conclusion:
The locus of the point \( P \) that divides the line segment \( AB \) into lengths \( a \) and \( b \) is an **ellipse**.