A HW | Cireles Questif Choose the equation of a eirele with radius 7 and center $$ \left(3_{1}=5\right) $$, Choose the eorrect answer below. A. $$ (x+3)^{\frac{3}{2}}+(y-5)^{2}=7 $$ B. $$ (x-3)^{2}+(y+5)^{2}=7 $$ C. $$ (x-3)^{2}+(y+5)^{2}=49 $$ D. $$ (x+3)^{2}+(y=5)^{2}=49 $$

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UpStudy AI · Accepted Answer

The standard equation of a circle with center $$(h, k)$$ and radius $$r$$ is:

$$
(x - h)^2 + (y - k)^2 = r^2
$$

Given:
- Center $$(3, 5)$$
- Radius $$7$$

Plugging these values into the formula:

$$
(x - 3)^2 + (y - 5)^2 = 7^2 \
(x - 3)^2 + (y - 5)^2 = 49
$$

Let's evaluate the provided options:

A. $$ (x + 3)^{\frac{3}{2}} + (y - 5)^2 = 7 $$  
- **Incorrect:** The exponent $$\frac{3}{2}$$ is wrong, and the $$x$$-term should be $$ (x - 3)^2 $$.

B. $$ (x - 3)^2 + (y + 5)^2 = 7 $$  
- **Incorrect:** The $$y$$-term should be $$ (y - 5)^2 $$, and the right side should be $$49$$.

C. $$ (x - 3)^2 + (y + 5)^2 = 49 $$  
- **Incorrect:** The $$y$$-term should be $$ (y - 5)^2 $$.

D. $$ (x + 3)^2 + (y = 5)^2 = 49 $$  
- **Incorrect:** The $$x$$-term should be $$ (x - 3)^2 $$, and the notation $$ (y = 5)^2 $$ is incorrect.

**None of the provided options are entirely correct.** However, option **C** is the closest to the correct equation, with the only mistake being the sign in the $$y$$-term. It should be $$ (y - 5)^2 $$ instead of $$ (y + 5)^2 $$.

**Correct Equation:**
$$
(x - 3)^2 + (y - 5)^2 = 49
$$
The correct equation of the circle is $$ (x - 3)^2 + (y - 5)^2 = 49 $$.

A HW | Cireles Questif Choose the equation of a eirele with radius 7 and center \( \left(3_{1}=5\right) \), Choose the eorrect answer below. A. \( (x+3)^{\frac{3}{2}}+(y-5)^{2}=7 \) B. \( (x-3)^{2}+(y+5)^{2}=7 \) C. \( (x-3)^{2}+(y+5)^{2}=49 \) D. \( (x+3)^{2}+(y=5)^{2}=49 \)

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