In a coordinate plane, how many points are both 5 units from the origin and 2 units from the x-axis?

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Answer 1 · 2017-04-25T22:38:23

4

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The loci of point 5 units from the origin is a circle of radius $5$ centred on the origin (red)

The loci of points 3 units from the $x$ -axis is two parallel lines $y=+-2$ (blue)

Visually we can see there are $4$ such points of intersection.

Proof

The loci of the circle is:

$x^2+y^2=25$

The loci of the two lines are:

$y=+-2$

At a point of intersection we have both equation satisfied simultaneously; thus:

$x^2+(+-2)^2=25$
$:. x^2+4=25$
$:. x^2=21$
$:. x=+-sqrt(21)$

Thus we have the four coordinates (all possible permutations) of:
coordinates $( +-sqrt(21), +-2)$ , ie:

$(-sqrt(21), 2), (-sqrt(21), -2), (sqrt(21), 2), (sqrt(21), -2)$