Math Problem Statement

Write the standard equation of a circle whose center

Solution

The image shows a handwritten problem asking to "Write the standard equation of the circle whose center is at (3,5)(-3, 5) and passes through the point (1,2)(1, 2)."

Solution:

  1. Identify the center and a point on the circle:

    • Center (h,k)=(3,5)(h, k) = (-3, 5)
    • Point on the circle (x1,y1)=(1,2)(x_1, y_1) = (1, 2)
  2. Formula for the standard equation of a circle: (xh)2+(yk)2=r2(x - h)^2 + (y - k)^2 = r^2 where rr is the radius.

  3. Find the radius: The radius rr is the distance between the center (h,k)(h, k) and the given point (x1,y1)(x_1, y_1), calculated by: r=(x1h)2+(y1k)2r = \sqrt{(x_1 - h)^2 + (y_1 - k)^2} Substituting the given values: r=(1(3))2+(25)2=(1+3)2+(3)2=42+(3)2=16+9=25=5r = \sqrt{(1 - (-3))^2 + (2 - 5)^2} = \sqrt{(1 + 3)^2 + (-3)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5

  4. Write the equation: Substitute h=3h = -3, k=5k = 5, and r=5r = 5 into the standard equation of the circle: (x(3))2+(y5)2=52(x - (-3))^2 + (y - 5)^2 = 5^2 Simplifying this: (x+3)2+(y5)2=25(x + 3)^2 + (y - 5)^2 = 25

Final Answer:

The standard equation of the circle is: (x+3)2+(y5)2=25(x + 3)^2 + (y - 5)^2 = 25

Would you like more details or have any questions?


Related Questions:

  1. What is the general form of a circle's equation?
  2. How do you find the radius if only the diameter is given?
  3. What are the steps to convert the general form to the standard form of a circle's equation?
  4. How can you find the center and radius from the general form of a circle's equation?
  5. What is the equation of a circle if its diameter endpoints are given?
  6. How does changing the center coordinates affect the circle's equation?
  7. How do you find the equation of a circle if it is tangent to the x-axis?
  8. How do you determine if a point lies inside, on, or outside a given circle?

Tip:

When given the center and a point on a circle, always start by finding the radius using the distance formula before substituting into the standard circle equation.

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Math Problem Analysis

Mathematical Concepts

Geometry
Circle Equations

Formulas

Distance Formula
Standard Equation of a Circle

Theorems

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Suitable Grade Level

Grades 9-12