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:
where is the center and is the radius.
Step 1: Find the center of the circle
The center is the midpoint of the diameter. The midpoint of the points and is calculated as:
Substituting the given points and :
So, the center is .
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:
Substituting the points and :
The radius is half of the diameter:
Step 3: Write the equation in standard form
Now, we can substitute the center and the radius into the standard form:
Simplifying the radius squared:
Thus, the equation of the circle is:
Would you like any further details or have any questions?
Here are 5 related questions:
- How do you find the equation of a circle given its center and a point on the circle?
- How do you convert the standard form of a circle equation to general form?
- How can you find the length of a diameter if you only know the radius?
- What is the relationship between a circle's diameter and circumference?
- 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
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