Question
\( 1 \leftarrow \begin{array}{l}\text { Write the standard form of the equation and the general form of the equation of the circle with } \\ \text { radius } r \text { and center }(h, k) \text {. Then graph the circle. } \\ r=1 ; \quad(h, k)=(8,6)\end{array} \) The standard form of the equation of this circle is (Type your answer in standard form.)
Ask by Bowers Kirk. in the United States
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The standard form of the circle's equation is \( (x - 8)^2 + (y - 6)^2 = 1 \).
Solution
The standard form of a circle’s equation with center (h, k) and radius r is given by:
(x – h)² + (y – k)² = r²
Plugging in h = 8, k = 6, and r = 1, we have:
(x – 8)² + (y – 6)² = 1
This is the standard form of the equation for the given circle.
To graph the circle:
1. Plot the center at (8, 6).
2. Since the radius is 1, mark points 1 unit above, below, left, and right of the center.
3. Draw a smooth curve connecting these points to form the circle.
The final answer in standard form is:
(x – 8)² + (y – 6)² = 1
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Simplify this solution The Deep Dive
The standard form of the equation of the circle with radius \( r = 1 \) and center \( (h, k) = (8, 6) \) is given by: \[ (x - 8)^2 + (y - 6)^2 = 1 \] The general form of the equation can be derived by expanding the standard form. This leads to: \[ x^2 - 16x + 64 + y^2 - 12y + 36 = 1 \] Simplifying this gives: \[ x^2 + y^2 - 16x - 12y + 99 = 0 \] To graph the circle, you would plot the center at the point (8, 6) and then draw a circle with a radius of 1, which extends one unit in all directions from the center. It will touch the points (9, 6), (7, 6), (8, 7), and (8, 5). Happy graphing!
