Question

A line of fixed length a + b, moves so that its ends are always on two fixed perpendicular straight lines. The locus of a point which divides the line into two parts of length a and b is

• A. A parabol
• B. A circle
• C. An ellips
• D. A hyperbol

Hint:

When a line segment divides into parts along fixed lines, the locus of such a point is typically an ellipse due to the constant sum of distances to the two fixed points (foci).

Answer 1

The Correct Option is C

Step 1: The problem describes a situation where a line segment of fixed length \(a + b\) moves, such that its endpoints always lie on two fixed perpendicular lines. A point divides the line segment into two parts, one of length \(a\) and the other of length \(b\). The task is to find the locus of this point.

Step 2: This geometric condition is characteristic of an ellipse. Specifically, the sum of the distances from any point on an ellipse to the two foci (the fixed points) is constant. In this case, the two fixed perpendicular lines act as the axes, and the point divides the line into parts \(a\) and \(b\), fulfilling the properties of an ellipse.

Step 3: Since the fixed points (the ends of the segment) maintain a constant distance relationship with the point dividing the segment, the locus of the point is an ellipse.