Question

Determine whether each point lies on the circle. Then use symmetry to determine another point on the circle.

Answer 1

Given:

Circle K has its center at (0,0) and a radius of 4.

Point N is at $(3,\sqrt{7})$.

To find:

To check whether the point N lies on the circle or not. Then find the other point using symmetry.

Solution:

The standard form of a circle is

$(x-h)^2+(y-k)^2=r^2$       ...(i)

where, (h,k) is center and r is radius.

Circle K has its center at (0,0) and a radius of 4.  Putting h=0, k=0 and r= 4 in (i), to get the equation of circle K.

$(x-0)^2+(y-0)^2=4^2$

$x^2+y^2=16$       ...(ii)

To check the point $N(3,\sqrt{7})$, put x=3 and $y=\sqrt{7}$ in (ii).

$(3)^2+(\sqrt{7})^2=16$

$9+7=16$  

$16=16$  

This statement is true. So. option $N(3,\sqrt{7})$ lies on the circle K.

According to the symmetry about the origin, if (x,y) lies on the graph then (-x,-y) also lies on that graph.

Since the center of the circle is origin, therefore, it is symmetrical about the origin.

Thus, point $P(-3,-\sqrt{7})$ must be on the circle K.

Therefore, the another point on the circle K is $P(-3,-\sqrt{7})$.