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Find the locus of the midpoint of the segment cut off by the coordinate axes from the line x cos α + y sin α = p, where p is a constant.

1. x² + y² = 4/p²
2. x² + y² = 4p²
3. 1/x² + 1/y² = 2/p²
4. 1/x² + 1/y² = 4/p²

Correct answer: 1/x² + 1/y² = 4/p²

Solution

The correct option represents the locus of the midpoint of the segment cut off by the coordinate axes from the given line. By finding the intercepts and calculating the midpoint, we derive the relationship that leads to the equation 1/x² + 1/y² = 4/p², which describes a rectangular hyperbola.

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