Math Problem Statement

Write the standard form of the equation of the circle with endpoints of a diameter at the points

left parenthesis 1 comma 2 right parenthesis(1,2)

and left parenthesis negative 3 comma 5 right parenthesis .

Solution

To find the standard form of the equation of a circle, we need to know its center and radius. The standard form of the equation is:

(xh)2+(yk)2=r2(x - h)^2 + (y - k)^2 = r^2

where (h,k)(h, k) is the center and rr is the radius.

Step 1: Find the center of the circle

The center is the midpoint of the diameter. The midpoint (h,k)(h, k) of the points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is calculated as:

h=x1+x22,k=y1+y22h = \frac{x_1 + x_2}{2}, \quad k = \frac{y_1 + y_2}{2}

Substituting the given points (1,2)(1, 2) and (3,5)(-3, 5):

h=1+(3)2=22=1h = \frac{1 + (-3)}{2} = \frac{-2}{2} = -1 k=2+52=72=3.5k = \frac{2 + 5}{2} = \frac{7}{2} = 3.5

So, the center is (1,3.5)(-1, 3.5).

Step 2: Find the radius

The radius is half the length of the diameter. The length of the diameter can be found using the distance formula:

d=(x2x1)2+(y2y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Substituting the points (1,2)(1, 2) and (3,5)(-3, 5):

d=((3)1)2+(52)2=(4)2+(3)2=16+9=25=5d = \sqrt{((-3) - 1)^2 + (5 - 2)^2} = \sqrt{(-4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5

The radius rr is half of the diameter:

r=52=2.5r = \frac{5}{2} = 2.5

Step 3: Write the equation in standard form

Now, we can substitute the center (1,3.5)(-1, 3.5) and the radius 2.52.5 into the standard form:

(x+1)2+(y3.5)2=(2.5)2(x + 1)^2 + (y - 3.5)^2 = (2.5)^2

Simplifying the radius squared:

(x+1)2+(y3.5)2=6.25(x + 1)^2 + (y - 3.5)^2 = 6.25

Thus, the equation of the circle is:

(x+1)2+(y3.5)2=6.25(x + 1)^2 + (y - 3.5)^2 = 6.25

Would you like any further details or have any questions?

Here are 5 related questions:

  1. How do you find the equation of a circle given its center and a point on the circle?
  2. How do you convert the standard form of a circle equation to general form?
  3. How can you find the length of a diameter if you only know the radius?
  4. What is the relationship between a circle's diameter and circumference?
  5. How do you find the center and radius of a circle from its general equation?

Tip: Always double-check calculations when using the distance formula, as small errors can lead to incorrect results.

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Math Problem Analysis

Mathematical Concepts

Geometry
Circles
Coordinate Geometry

Formulas

Midpoint formula: (h, k) = ((x1 + x2)/2, (y1 + y2)/2)
Distance formula: d = √((x2 - x1)^2 + (y2 - y1)^2)
Standard form of a circle: (x - h)^2 + (y - k)^2 = r^2

Theorems

Midpoint Theorem
Pythagorean Theorem

Suitable Grade Level

Grades 9-11