Question

The locus of the midpoint of the portion of the line \( x \cos \alpha + y \sin \alpha = p \) intercepted by the coordinate axes, where \( p \) is a constant, is:

• A. \( \frac{1}{x^2} + \frac{1}{y^2} = \frac{3}{p^2} \)
• B. \( \frac{1}{x^2} + \frac{1}{y^2} = \frac{4}{p^2} \)
• C. \( x^2 + y^2 = 2p^2 \)
• D. \( \frac{2}{x^2} + \frac{1}{p^2} = 1 \)

Hint:

In geometry problems involving loci, first find the intercepts and then calculate the midpoint of the intercepted line segment.

Answer 1

The Correct Option is B

The equation of the line is \( x \cos \alpha + y \sin \alpha = p \). We are interested in finding the locus of the midpoint of the portion of the line intercepted by the coordinate axes.
- When \( y = 0 \), \( x = \frac{p}{\cos \alpha} \).
- When \( x = 0 \), \( y = \frac{p}{\sin \alpha} \). The midpoint \( M \) of the segment connecting these intercepts is the average of the intercepts: \[ M = \left( \frac{p}{2 \cos \alpha}, \frac{p}{2 \sin \alpha} \right). \] The locus of the midpoint is given by the equation: \[ \frac{1}{x^2} + \frac{1}{y^2} = \frac{4}{p^2}. \]