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) #