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

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

Solution

failed
failed

Solution Steps

To find the intercepts of the given equation \(x^2 + 25y^2 = 25\), we need to determine where the graph intersects the x-axis and y-axis. For x-intercepts, set \(y = 0\) and solve for \(x\). For y-intercepts, set \(x = 0\) and solve for \(y\). To test for symmetry, check if the equation is symmetric with respect to the x-axis, y-axis, and the origin by substituting \((-x, y)\), \((x, -y)\), and \((-x, -y)\) into the equation and verifying if the equation remains unchanged.

Step 1: Finding the x-Intercepts

To find the x-intercepts, we set \(y = 0\) in the equation \(x^2 + 25y^2 = 25\). This simplifies to: \[ x^2 = 25 \] Solving for \(x\), we find: \[ x = \pm 5 \] Thus, the x-intercepts are \((-5, 0)\) and \((5, 0)\).

Step 2: Finding the y-Intercepts

Next, we find the y-intercepts by setting \(x = 0\) in the equation \(x^2 + 25y^2 = 25\). This simplifies to: \[ 25y^2 = 25 \] Solving for \(y\), we get: \[ y^2 = 1 \quad \Rightarrow \quad y = \pm 1 \] Therefore, the y-intercepts are \((0, -1)\) and \((0, 1)\).

Step 3: Testing for Symmetry

To test for symmetry:

  • Symmetry with respect to the x-axis: Substituting \((x, -y)\) into the equation yields the same equation, indicating symmetry.
  • Symmetry with respect to the y-axis: Substituting \((-x, y)\) also yields the same equation, indicating symmetry.
  • Symmetry with respect to the origin: Substituting \((-x, -y)\) yields the same equation, indicating symmetry.

Final Answer

The intercepts are:

  • x-intercepts: \((-5, 0)\) and \((5, 0)\)
  • y-intercepts: \((0, -1)\) and \((0, 1)\)

The equation is symmetric with respect to the x-axis, y-axis, and the origin.

Thus, the final answer is: \[ \boxed{\text{x-intercepts: } (-5, 0), (5, 0) \text{; y-intercepts: } (0, -1), (0, 1)} \]

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
Which of these describes humoral immunity? - T-cells and macrophages fighting infected cells that are non-self or cancerous - obtaining immunity from an outside source, like mother to baby - B-cells binding to antigens to create a response of plasma cells and memory b-cells
failedAnswered
Write the formula for the ionic compound formed from Zn^2+ and each anion given below. Then give the name of each compound. Part 1 of 2 OH^- formula: name: Part 2 of 2
failedAnswered
Which of the following is most likely to be asset price inflation? A rise in the current price of assets caused by expectation of higher asset prices in the future A rise in the price of assets caused by an increase in the underlying productivity of those assets A rise in the current price of assets caused by a shift in people's time preference A rise in the current price of assets caused by an increase in the riskiness of financial assets
failedAnswered
Question 3 of 50 When molecules absorb heat, there is an increase in A the kinetic energy of the molecules. B The potential energy of the molecules. C The mass of the molecules. D The bond energy of the molecules.
failedAnswered
The refusal of a lending institution to make a residential real estate loan strictly because of the racial or ethnic composition of the neighborhood is called Select one: a. blockbusting. b. panic peddling. C. steering. d. redlining.