Question: A line segment of length 2a slides with its ends on the x and y axes. Find the polar equation of the locus described by the point P(r,θ) in which the perpendicular from the origin intersects the moving line segment.
Solution: The line segment with its ends sliding along the axes (pictured red in this diagram) forms the hypotenuse of a right triangle. I'll call this red line segment a "stick". Let "A" represent the y-axis end of the stick, and O the origin. Angle A has measure θ, so AO has length 2a cos θ. We'll let "B" represent the x-axis end of the stick, so BO has length 2a sin θ.
The blue line is perpendicular to the stick, so it is an altitude of triangle AOB, and its angle with horizontal is θ. As you no doubt know, an altitude (to the hypotenuse) of a right triangle divides the triangle into two right triangles both similar to the larger triangle.
Since triangle OPB is similar to triangle AOB, we know the length OP is the length OB times cos θ, so
OP = 2a sin θ cos θ
OP = a sin 2θ.
Thus the locus of point P is given by the polar equation,
r = a sin 2θ,
which is the equation of the quadrifolium.
Back to Polar Coordinates, or all the way back to Geometry and Trig
The quadrifolium is defined in the Geometry Glossary.
The webmaster and author of this Math Help site is Graeme McRae.