Problem 22 · 2015 Math Kangaroo
Stretch
Geometry & Measurement
symmetry
The curve in the diagram is defined by the equation (x2 + y2 − 2x)2 = 2(x2 + y2). Which of the lines a, b, c, d is the y-axis?
Show answer
Answer: A — a
Show hints
Hint 1 of 2
Rewrite the curve in polar form to see its single axis of symmetry.
Still stuck? Show hint 2 →
Hint 2 of 2
The y-axis must be perpendicular to that axis of symmetry, then match it to the drawn lines.
Show solution
Approach: find the curve's symmetry axis and locate the perpendicular line
- Substituting \(x^2+y^2=r^2\) and \(x=r\cos\theta\), the equation becomes \(r = 2\cos\theta \pm \sqrt2\); since it depends only on \(\cos\theta\), the limaçon is symmetric about its own x-axis, with the small inner loop opening toward the positive-x side.
- The y-axis passes through the curve's origin (its self-crossing node) and is perpendicular to that symmetry axis.
- Matching the perpendicular-through-the-node line to the drawing identifies it as a (A).
Mark:
· log in to save