Consider the locus of a moving point P = $(x, y)$ in the plane which satisfies the law $2x^2 = r^2 + r^4$, where $r^2 = x^2 + y^2$. Then only one of the following statements is true. Which one is it?
(a) For every positive real number d, there is a point $(x, y)$ on the locus such that $r = d$.
(b) For every value $d$, $0 < d < 1$, there are exactly four points on the locus, each of which is at a distance $d$ from the origin.
(c) The point P always lies in the first quadrant.
(d) The locus of P is an ellipse.
The answer is option b.
I could eliminate options c and d because it's not an ellipse and $(x,y)$ can be in any quadrant. However, I'm struggling to disprove option 'a'. How do I do that?
Best Answer
(b) Given $0<d<1$, we find exactly two solutions for $x$ in $2x^2=d^2+d^4$ because the right hand side is positive. Likewise, we find two solutions for $y$ in $2y^2=2(d^2-x^2)=d^2-d^4$, again because the right hand side is positive. By construction, we have $x^2+y^2=d^2$ in all four combined cases.
(a) Note that we found $2y^2=d^2-d^4$ in the proof for (b). For $d>1$ the right hand side is negative.