Find the sum of the following infinite geometric series, if it exists.
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Answer:
The coordinates of D are (8,3)
Step-by-step explanation:
We have quadrilateral ABCD being a parallelogram with coordinates A(3,2), B(5,4), C(10,5), and D(a,b).
We want to find the coordinates of D.
Since ABCD is a parallelogram, the diagonals AC and BD bisects each other.
The midpoint of AC is (13/2,7/2).
This is also the midpoint of BD.
By the midpoint rule:
This implies that:
Similarly,
The coordinates of D are (8,3)
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: