The problem asks to find the "locus of P(x, y)" based on the geometric setup in the image. Here’s a step-by-step explanation:

Description of the Diagram:

• The diagram shows a right triangle, with the point $P(x, y)$ being a movable point on the hypotenuse.
• The horizontal distance is labeled $a$, and the vertical distance is labeled $b$.
• The hypotenuse has a length $m$, connecting the origin $(0, 0)$ to $P(x, y)$.

Interpretation and Problem Objective:

The problem seems to be asking for the equation or locus of the point $P(x, y)$, given the constraints of the triangle.

Since $P(x, y)$ lies on the hypotenuse, we can apply the Pythagorean Theorem. The equation that relates the sides of the right triangle is:

$a^2 + b^2 = m^2$

This suggests that the point $P(x, y)$ lies on a circle centered at the origin with radius $m$.

Conclusion:

The locus of $P(x, y)$ is the set of all points at a constant distance $m$ from the origin. This is the equation of a circle. Therefore, the equation of the locus is:

$x^2 + y^2 = m^2$

This is the equation of a circle with radius $m$ and center at $(0, 0)$.

Do you want further details or have any questions?

Related Questions:

1. How can we derive the equation of the locus geometrically?
2. What if $P(x, y)$ were on another line, how would the locus change?
3. What are the parametric equations for this circle?
4. How can we verify that the point $P(x, y)$ lies on the circle?
5. Can this problem be extended to higher dimensions?

Tip:

Always remember that when a point moves while maintaining a constant distance from another fixed point, its locus is typically a circle (in 2D geometry).