For the given equation, list the intercepts and test for symmetry. $$ x^{2}+25 y^{2}=25 $$ What are the intercept(s)? Select the correct choice below and fill in any answer boxes within your choice. A. The intercept(s) is/are $$ \square $$. (Type an ordered pair. Use a comma to separate answers as needed.) B. There are no intercepts.

Question

UpStudy AI · Accepted Answer

Function by following steps:
- step0: Find the $$x$$-intercept/zero:
  $$y=\frac{\sqrt{25-x^{2}}}{5}$$
- step1: Set $$y$$=0$$:$$
  $$0=\frac{\sqrt{25-x^{2}}}{5}$$
- step2: Swap the sides:
  $$\frac{\sqrt{25-x^{2}}}{5}=0$$
- step3: Find the domain:
  $$\frac{\sqrt{25-x^{2}}}{5}=0,-5\leq x\leq 5$$
- step4: Simplify:
  $$\sqrt{25-x^{2}}=0$$
- step5: Set the radicand equal to 0:
  $$25-x^{2}=0$$
- step6: Move the constant to the right side:
  $$-x^{2}=0-25$$
- step7: Remove 0:
  $$-x^{2}=-25$$
- step8: Change the signs:
  $$x^{2}=25$$
- step9: Simplify the expression:
  $$x=\pm \sqrt{25}$$
- step10: Simplify:
  $$x=\pm 5$$
- step11: Separate into possible cases:
  $$\begin{align}&x=5\&x=-5\end{align}$$
- step12: Check if the solution is in the defined range:
  $$\begin{align}&x=5\&x=-5\end{align},-5\leq x\leq 5$$
- step13: Find the intersection:
  $$\begin{align}&x=5\&x=-5\end{align}$$
- step14: Calculate:
  $$x_{1}=-5,x_{2}=5$$
Analyze the x-Axis symmetry of the function $$ f(x)=\sqrt{\frac{25-x^{2}}{25}} $$
Testing for symmetry by following steps:
- step0: Testing for symmetry about the x-axis:
  $$f\left(x\right)=\frac{\sqrt{25-x^{2}}}{5}$$
- step1: Use the appropriate notation:
  $$y=\frac{\sqrt{25-x^{2}}}{5}$$
- step2: Use substitution:
  $$-y=\frac{\sqrt{25-x^{2}}}{5}$$
- step3: Change the signs both sides:
  $$y=-\frac{\sqrt{25-x^{2}}}{5}$$
- step4: The graph of $$y=\frac{\sqrt{25-x^{2}}}{5}$$ is not symmetry with respect to the x-axis$$:$$
  $$\textrm{Not symmetry with respect to the x-axis}$$
Analyze the y-Axis symmetry of the function $$ f(x)=\sqrt{\frac{25-x^{2}}{25}} $$
Testing for symmetry by following steps:
- step0: Testing for symmetry about the y-axis:
  $$f\left(x\right)=\frac{\sqrt{25-x^{2}}}{5}$$
- step1: Use the appropriate notation:
  $$y=\frac{\sqrt{25-x^{2}}}{5}$$
- step2: Use substitution:
  $$y=\frac{\sqrt{25-\left(-x\right)^{2}}}{5}$$
- step3: Simplify:
  $$y=\frac{\sqrt{25-x^{2}}}{5}$$
- step4: The graph of $$y=\frac{\sqrt{25-x^{2}}}{5}$$ is symmetry with respect to the y-axis$$:$$
  $$\textrm{Symmetry with respect to the y-axis}$$
Analyze the origin symmetry of the function $$ f(x)=\sqrt{\frac{25-x^{2}}{25}} $$
Testing for symmetry by following steps:
- step0: Testing for symmetry about the origin:
  $$f\left(x\right)=\frac{\sqrt{25-x^{2}}}{5}$$
- step1: Use the appropriate notation:
  $$y=\frac{\sqrt{25-x^{2}}}{5}$$
- step2: Use substitution:
  $$-y=\frac{\sqrt{25-\left(-x\right)^{2}}}{5}$$
- step3: Simplify:
  $$-y=\frac{\sqrt{25-x^{2}}}{5}$$
- step4: Change the signs both sides:
  $$y=-\frac{\sqrt{25-x^{2}}}{5}$$
- step5: The graph of $$y=\frac{\sqrt{25-x^{2}}}{5}$$ is not symmetry with respect to the origin$$:$$
  $$\textrm{Not symmetry with respect to the origin}$$
Analyze the y intercept of the function $$ y=\sqrt{\frac{25-x^{2}}{25}} $$
Function by following steps:
- step0: Find the y-intercept:
  $$y=\frac{\sqrt{25-x^{2}}}{5}$$
- step1: Set $$x$$=0$$:$$
  $$y=\frac{\sqrt{25-0^{2}}}{5}$$
- step2: Simplify:
  $$y=1$$
Let's analyze the equation $$ x^{2}+25 y^{2}=25 $$ to find the intercepts and test for symmetry.

### Step 1: Finding the Intercepts

**X-Intercepts:**
To find the x-intercepts, we set $$ y = 0 $$ in the equation:

$$
x^{2} + 25(0)^{2} = 25 \implies x^{2} = 25 \implies x = \pm 5
$$

Thus, the x-intercepts are:
$$
(-5, 0) \quad \text{and} \quad (5, 0)
$$

**Y-Intercept:**
To find the y-intercept, we set $$ x = 0 $$ in the equation:

$$
(0)^{2} + 25y^{2} = 25 \implies 25y^{2} = 25 \implies y^{2} = 1 \implies y = \pm 1
$$

Thus, the y-intercepts are:
$$
(0, 1) \quad \text{and} \quad (0, -1)
$$

### Summary of Intercepts
The intercepts are:
- X-Intercepts: $$(-5, 0), (5, 0)$$
- Y-Intercepts: $$(0, 1), (0, -1)$$

### Step 2: Testing for Symmetry

1. **Symmetry with respect to the y-axis:**
   - The equation is symmetric with respect to the y-axis because replacing $$ x $$ with $$ -x $$ does not change the equation.

2. **Symmetry with respect to the x-axis:**
   - The equation is not symmetric with respect to the x-axis because replacing $$ y $$ with $$ -y $$ changes the equation.

3. **Symmetry with respect to the origin:**
   - The equation is not symmetric with respect to the origin because replacing $$ x $$ with $$ -x $$ and $$ y $$ with $$ -y $$ changes the equation.

### Final Answer
A. The intercept(s) is/are $$(-5, 0), (5, 0), (0, 1), (0, -1)$$.
The intercepts are $$(-5, 0)$$, $$(5, 0)$$, $$(0, 1)$$, and $$(0, -1)$$.

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Pre Calculus Questions