A Circle Has A Center At 4 − 5 I 4-5i 4 − 5 I And A Point On The Circle At 19 − 13 I 19-13i 19 − 13 I . Which Of The Following Points Is Also On The Circle?A. − 11 − 3 I -11-3i − 11 − 3 I B. − 4.5 + 3.5 I -4.5+3.5i − 4.5 + 3.5 I C. 12 + 10 I 12+10i 12 + 10 I D. 21 + 12 I 21+12i 21 + 12 I

Introduction

In the complex plane, a circle can be defined as the set of all points that are a fixed distance from a given center point. This distance is known as the radius of the circle. Given the center and a point on the circle, we can use the concept of the distance formula to find other points that lie on the circle. In this article, we will explore how to find a point on a circle in the complex plane using the distance formula.

The Distance Formula in the Complex Plane

The distance formula in the complex plane is a generalization of the distance formula in the Cartesian plane. Given two complex numbers $z_1 = a + bi$ and $z_2 = c + di$, the distance between them is given by:

$$d = |z_2 - z_1| = \sqrt{(c - a)^2 + (d - b)^2}$$

where $|z|$ denotes the modulus (or magnitude) of the complex number $z$.

The Center and a Point on the Circle

We are given that the center of the circle is at $4 - 5i$ and a point on the circle is at $19 - 13i$. We want to find another point that lies on the circle. To do this, we will use the distance formula to find the radius of the circle.

Finding the Radius of the Circle

The radius of the circle is the distance between the center and the point on the circle. Using the distance formula, we have:

$$r = |19 - 13i - (4 - 5i)| = |15 - 8i| = \sqrt{15^2 + (-8)^2} = \sqrt{225 + 64} = \sqrt{289} = 17$$

Finding a Point on the Circle

Now that we have the radius of the circle, we can use the distance formula to find another point that lies on the circle. We want to find a point $z = x + yi$ such that the distance between the center and $z$ is equal to the radius.

$$|z - (4 - 5i)| = 17$$

We can rewrite this equation as:

$$\sqrt{(x - 4)^2 + (y + 5)^2} = 17$$

Squaring both sides, we get:

$$(x - 4)^2 + (y + 5)^2 = 289$$

Evaluating the Options

We are given four options for the point $z = x + yi$. We can substitute each option into the equation above and check if it satisfies the equation.

Option A: $-11 - 3i$

Substituting $x = -11$ and $y = -3$ into the equation, we get:

$$(x - 4)^2 + (y + 5)^2 = (-11 - 4)^2 + (-3 + 5)^2 = (-15)^2 + 2^2 = 225 + 4 = 229$$

This does not satisfy the equation.

Option B: $-4.5 + 3.5i$

Substituting $x = -4.5$ and $y = 3.5$ into the equation, we get:

$$(x - 4)^2 + (y + 5)^2 = (-4.5 - 4)^2 + (3.5 + 5)^2 = (-8.5)^2 + 8.5^2 = 72.25 + 72.25 = 144.5$$

This does not satisfy the equation.

Option C: $12 + 10i$

Substituting $x = 12$ and $y = 10$ into the equation, we get:

$$(x - 4)^2 + (y + 5)^2 = (12 - 4)^2 + (10 + 5)^2 = 8^2 + 15^2 = 64 + 225 = 289$$

This satisfies the equation.

Option D: $21 + 12i$

Substituting $x = 21$ and $y = 12$ into the equation, we get:

$$(x - 4)^2 + (y + 5)^2 = (21 - 4)^2 + (12 + 5)^2 = 17^2 + 17^2 = 289 + 289 = 578$$

This does not satisfy the equation.

Conclusion

We have found that the point $12 + 10i$ lies on the circle. This is the correct answer.

Final Answer

The final answer is $\boxed{C}$

Introduction

In our previous article, we explored how to find a point on a circle in the complex plane using the distance formula. We used the concept of the distance formula to find the radius of the circle and then used it to find another point that lies on the circle. In this article, we will answer some frequently asked questions about circles in the complex plane.

Q: What is the center of the circle?

A: The center of the circle is the point from which all points on the circle are equidistant. In the complex plane, the center of the circle is represented by a complex number.

Q: How do I find the radius of the circle?

A: To find the radius of the circle, you need to find the distance between the center and a point on the circle. You can use the distance formula to do this.

Q: How do I find a point on the circle?

A: To find a point on the circle, you need to find a point that is equidistant from the center. You can use the distance formula to find this point.

Q: What is the equation of a circle in the complex plane?

A: The equation of a circle in the complex plane is given by:

$$(x - a)^2 + (y - b)^2 = r^2$$

where $(a, b)$ is the center of the circle and $r$ is the radius.

Q: How do I find the equation of a circle in the complex plane?

A: To find the equation of a circle in the complex plane, you need to know the center and the radius of the circle. You can use the distance formula to find the radius and then use the equation above to find the equation of the circle.

Q: Can I find the equation of a circle in the complex plane if I only know a point on the circle?

A: Yes, you can find the equation of a circle in the complex plane if you only know a point on the circle. You can use the distance formula to find the radius and then use the equation above to find the equation of the circle.

Q: How do I find the intersection of two circles in the complex plane?

A: To find the intersection of two circles in the complex plane, you need to find the points that are common to both circles. You can use the equation of the circles to find these points.

Q: Can I find the intersection of two circles in the complex plane if I only know the centers and radii of the circles?

A: Yes, you can find the intersection of two circles in the complex plane if you only know the centers and radii of the circles. You can use the equation of the circles to find the intersection points.

Q: How do I find the distance between two points in the complex plane?

A: To find the distance between two points in the complex plane, you can use the distance formula.

Q: Can I find the distance between two points in the complex plane if I only know the complex numbers representing the points?

A: Yes, you can find the distance between two points in the complex plane if you only know the complex numbers representing the points. You can use the distance formula to find the distance.

Conclusion

We have answered some frequently asked questions about circles in the complex plane. We hope this article has been helpful in understanding the concepts of circles in the complex plane.

Final Answer

The final answer is that the distance formula is a powerful tool for finding points on a circle in the complex plane, and the equation of a circle in the complex plane is given by $(x - a)^2 + (y - b)^2 = r^2$.