Question

A circle has a center at 4 – 5i and a point on the circle at 19 – 13i. Which of the following points is also on the circle?

–11 –3i
–4.5 +3.5i
12 + 10i
21 + 12i

Answer 1

Hers is a solution : 

Answer 2

The correct option is 3.

Step-by-step explanation:

The center of the circle is (4-5i). A point on the circle at (19 – 13i).

Distance formula between two points
`z_1=x_1+iy_1\text{ and } z_2=x_2+iy_2` is

`d=√((x_2-x_1)^2+(y_2-y_1)^2)`

The radius of the circle is

`d=√((19-4)^2+(-13-(-5))^2)=17`

The radius of the circle is 17 units.

The distance between –11 –3i and (4-5i).

`d=√((-11-4)^2+(-4-(-5))^2)=√(226)`

The distance between –4.5+3.5i and (4-5i).

`d=√((-4.5-4)^2+(3.5-(-5))^2)=12.02`

The distance between -12+10i and (4-5i).

`d=√((12-4)^2+(10-(-5))^2)=17`

The distance between -12+10i and (4-5i) is equal to the radius. Therefore point 12+10i lies on the circle.

The distance between 21+12i and (4-5i).

`d=√((21-4)^2+(12-(-5))^2)=21.93`

Therefore option 3 is correct.