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

To solve the problem we must know about the equation of a circle and complex numbers.

Equation of circle

Radius² = (Distance between the center and any point on the circle)²

Radius = $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Complex Number

Complex Number, z = (x +iy)

Another point that will lay on the same circle is (12 + 10i).

Explanation

Given to us

• center = (4 – 5i)
• point on the circle = (19 – 13i)

Solution

We know that equation for a circle can be written as,

Radius of the Circle

Radius of the Circle

= √(Distance between the center and any point on the circle)²

Radius of the Circle = √[4-19]²+[-5 -(-13)]²

                                 = √[-15]²+[8]²

                                 = √225 + 64

                                 = √225 + 64

                                 = √289

                                 = 17

Now compare the distance between the center of the circle and the points.

Comparison

A.)  –11 –3i

Distance between the center and point on the circle

= √[4-(-11)]² + [-5 - (-3)]²

= √[15]² + [-2]²

= √225 + 4

= √229

B.)  –4. 5+3. 5i

Distance between the center and point on the circle

= √[4-(-4.5)]² + [-5 - (3.5)]²

= √[9.5]² + [-8.5]²

= √90.25+ 72.25

= √162.5

C.)  12 + 10i

Distance between the center and point on the circle

= √[4-(12)]² + [-5 - (10)]²

= √[-8]² + [-15]²

= √64+ 225

= √289

= 17

D.)  21 + 12i.

Distance between the center and point on the circle

= √[4-(21)]² + [-5 - (12)]²

= √[17]² + [-17i]²

= √289+ 289

= √578

Hence, another point that will lay on the same circle is (12 + 10i).

Learn more about Equation of Circle:

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