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Which of the following is a result of shifting a circle with the equation [tex](x+3)^2+(y-2)^2=36[/tex] left 3 units?

A. Both the [tex]$x$[/tex]- and [tex]$y$[/tex]-coordinates of the center of the circle decrease by 3.
B. The [tex]$x$[/tex]-coordinate of the center of the circle decreases by 3.
C. The [tex]$y$[/tex]-coordinate of the center of the circle increases by 3.
D. The [tex]$x$[/tex]-coordinate of the center of the circle increases by 3.

Sagot :

GinnyAnswer
To solve the given problem, we need to determine the effect of shifting the circle described by the equation [tex]\((x+3)^2 + (y-2)^2 = 36\)[/tex] left by 3 units.

### Step-by-Step Solution:

1. Identify the center and radius of the given circle:
The equation of a circle in standard form is [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\((h, k)\)[/tex] is the center and [tex]\(r\)[/tex] is the radius.
- Given equation: [tex]\((x + 3)^2 + (y - 2)^2 = 36\)[/tex]
- This can be rewritten as [tex]\((x - (-3))^2 + (y - 2)^2 = 36\)[/tex].
- Therefore, the center of the circle is [tex]\((-3, 2)\)[/tex], and the radius is [tex]\(\sqrt{36} = 6\)[/tex].

2. Determine the effect of shifting the circle left by 3 units:
- Shifting the circle left by 3 units means subtracting 3 from the [tex]\(x\)[/tex]-coordinate of the center.
- Original [tex]\(x\)[/tex]-coordinate of the center: [tex]\(-3\)[/tex]
- New [tex]\(x\)[/tex]-coordinate of the center after shifting left by 3 units: [tex]\(-3 - 3 = -6\)[/tex]

3. Check the [tex]\(y\)[/tex]-coordinate:
- The [tex]\(y\)[/tex]-coordinate remains unaffected when the circle is shifted left.
- Original [tex]\(y\)[/tex]-coordinate of the center: [tex]\(2\)[/tex]

4. New coordinates of the center:
- After the shift, the new coordinates of the center are [tex]\((-6, 2)\)[/tex].

5. Verify the correct option:
- Option A: Both the [tex]\(x\)[/tex]- and [tex]\(y\)[/tex]-coordinates of the center of the circle decrease by 3. (Incorrect, only the [tex]\(x\)[/tex]-coordinate changes)
- Option B: The [tex]\(x\)[/tex]-coordinate of the center of the circle decreases by 3. (Correct)
- Option C: The [tex]\(y\)[/tex]-coordinate of the center of the circle increases by 3. (Incorrect, the [tex]\(y\)[/tex]-coordinate does not change)
- Option D: The [tex]\(x\)[/tex]-coordinate of the center of the circle increases by 3. (Incorrect, the [tex]\(x\)[/tex]-coordinate decreases)

Thus, the correct result of shifting the circle left by 3 units is:
B. The [tex]\(x\)[/tex]-coordinate of the center of the circle decreases by 3.
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