Select The Correct Answer.Which Point Lies On The Circle Represented By The Equation \[$(x+5)^2+(y-9)^2=8^2?\$\]A. \[$(0,8)\$\] B. \[$(13,-9)\$\] C. \[$(-5,1)\$\] D. \[$(3,17)\$\]

The equation of a circle in standard form is given by ${(x-h)^2+(y-k)^2=r^2,}$ where ${(h,k)}$ represents the center of the circle and ${r}$ is the radius. In the given equation {(x+5)^2+(y-9)2=8^2,$] we can identify the center of the circle as ${(-5,9)}$ and the radius as ${8}$.

Analyzing the Options

To determine which point lies on the circle, we need to substitute each option into the equation and check if it satisfies the equation.

Option A: [(0,8)}$

Substituting ${(0,8)}$ into the equation, we get:

${(0+5)^2+(8-9)^2=8^2}$ ${25+1=64}$ ${26\neq64}$

Since the equation does not hold true for option A, we can eliminate it.

Option B: ${(13,-9)}$

Substituting ${(13,-9)}$ into the equation, we get:

${(13+5)^2+(-9-9)^2=8^2}$ ${18^2+(-18)^2=8^2}$ ${324+324=64}$ ${648\neq64}$

Since the equation does not hold true for option B, we can eliminate it.

Option C: ${(-5,1)}$

Substituting ${(-5,1)}$ into the equation, we get:

${(-5+5)^2+(1-9)^2=8^2}$ ${0^2+(-8)^2=8^2}$ ${0+64=64}$

Since the equation holds true for option C, we can conclude that the point ${(-5,1)}$ lies on the circle.

Option D: ${(3,17)}$

Substituting ${(3,17)}$ into the equation, we get:

${(3+5)^2+(17-9)^2=8^2}$ ${8^2+8^2=8^2}$ ${64+64=64}$ ${128\neq64}$

Since the equation does not hold true for option D, we can eliminate it.

Conclusion

In this article, we will address some of the most frequently asked questions about circles, including their equations, properties, and applications.

Q: What is the standard form of the equation of a circle?

A: The standard form of the equation of a circle is given by [(x-h)^2+(y-k)2=r^2,}$ where ${(h,k)}$ represents the center of the circle and ${r}$ is the radius.

Q: How do I determine the center and radius of a circle from its equation?

A: To determine the center and radius of a circle from its equation, you need to rewrite the equation in standard form. The center of the circle will be represented by the values of ${h}$ and ${k}$, and the radius will be represented by the value of ${r}$.

Q: What is the difference between the center and the radius of a circle?

A: The center of a circle is the point around which the circle is centered, while the radius is the distance from the center to any point on the circle.

Q: How do I find the distance between two points on a circle?

A: To find the distance between two points on a circle, you can use the distance formula: ${d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.}$

Q: What is the circumference of a circle?

A: The circumference of a circle is given by the formula: ${C=2\pi r,}$ where ${r}$ is the radius of the circle.

Q: What is the area of a circle?

A: The area of a circle is given by the formula: ${A=\pi r^2,}$ where ${r}$ is the radius of the circle.

Q: How do I graph a circle?

A: To graph a circle, you need to plot the center of the circle and then draw a circle with a radius equal to the distance from the center to any point on the circle.

Q: What are some real-world applications of circles?

A: Circles have many real-world applications, including:

• Geometry and trigonometry
• Architecture and engineering
• Art and design
• Physics and astronomy
• Computer science and programming

Q: How do I use circles in real-world problems?

A: To use circles in real-world problems, you need to apply the concepts and formulas of circle geometry to solve problems in fields such as architecture, engineering, art, and science.

Conclusion

In this article, we have addressed some of the most frequently asked questions about circles, including their equations, properties, and applications. We hope that this information has been helpful in understanding the concept of circles and how to apply it in real-world problems.