Question

# Prove or disprove that the point (1, sqrt3) lies on the circle that is centered at the origin and contains the point (0,2)

Circles

Prove or disprove that the point (1, $$\sqrt{3}$$) lies on the circle that is centered at the origin and contains the point (0,2)

2021-03-09

The standard equation of a circle with center (h,k) and radius r is given by:
$$\displaystyle{\left({x}−{h}\right)}^{{2}}+{\left({y}−{k}\right)}^{{2}}={r}^{{2}}$$
Using $$(h,k)=(0,0)$$ and $$(x,y)=(0,2)$$, we solve for r2:
$$(0-0)2+(2-0)2=r2$$
$$4=r2$$
So, the equation of the circle is:
$$\displaystyle{x}^{{2}}+{y}^{{2}}={4}$$
The point (1,$$\sqrt{3}$$) lies on the circle if it satisfies the equation so we check:
$$\displaystyle{1}^{{2}}+{\left(√{3}\right)}^{{2}}=?{4}$$
$$1+3=?4$$
$$4=4$$
Since it satisfies the equation, then $$\displaystyle{\left({1},√{3}\right)}$$ lies on the circle.