True or False? #4

Algebra Level 5

Which of these followings are true?

A. The graph of $xy=0$ is a union of the $x$ -axis and the $y$ -axis.

B. If the graph of $x^2+y^2=r$ appears on the Cartesian coordinates system, then $r>0.$

C. The graph of $|y|=\sqrt{-2x^2+3xy}$ and $x = c$ intersects at two points for all real values $c\neq 0$ .

This problem is a part of <True or False?> series .

Only C None of them Only A Only B and C Only B Only C and A Only A and B All of them

1 solution

Boi (보이)
Aug 19, 2017

$xy=0$ means $x=0$ or $y=0,$ which are the $y$ -axis and $x$ -axis respectively.

$\therefore~\boxed{\text{TRUE}}$

$x^2+y^2=0$ appears on the Cartesian coordinates system as $(0,~0),$ so $r\ge0.$

$\therefore~\boxed{\text{FALSE}}$

Square both sides, and you get $y^2=-2x^2+3xy,$ or $(y-x)(y-2x)=0.$

This means $y=x$ or $y=2x,$ and $x=c$ is parallel to the $y$ -axis, and not the same with $y$ -axis.

The graph shows $y=x$ and $y=2x$ merged together.

$\therefore~\boxed{\text{TRUE}}$

From above, we see that only A and C are true.

Wow, this question tricked me because I thought the equation in C was an equation of an ellipse when it was in fact a degenerate conic section.

James Wilson - 3 years, 9 months ago