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To solve the problem we must know about the equation of a circle and complex numbers.
<h2>Equation of circle</h2>Radius² = (Distance between the center and any point on the circle)²
Radius =
Complex Number, z = (x +iy)
Another point that will lay on the same circle is (12 + 10i).
<h2>Explanation</h2>Given to us
- center = (4 – 5i)
- point on the circle = (19 – 13i)
Solution
We know that equation for a circle can be written as,
<h3>Radius of the Circle </h3>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.
<h3>Comparison</h3>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:
Given:
To find:
The value of .
Solution:
We know that,
...(i)
Where n is an integer.
We have,
Using (i), it can be written as
It is given that .
Therefore, the value of the given function expression is .