Math Problem Statement
Solution
Let's break down and solve the problem based on the equation given in the image:
The given equation is: This equation describes an ellipse, and we need to solve the questions related to it.
Step 1: Rewrite the equation of the ellipse
We start by grouping the and terms to complete the square:
Now, factor out the coefficients of and from the grouped terms:
Step 2: Complete the square for and
For , take half of 14 (which is 7) and square it (49), then add and subtract 49 inside the parentheses:
= 144((x + 7)^2 - 49)$$ Similarly, for $$y^2 - 8y$$, take half of 8 (which is 4) and square it (16), then add and subtract 16 inside the parentheses: $$81(y^2 - 8y + 16 - 16) = 81((y - 4)^2 - 16)$$ ### Step 3: Substitute the completed squares Now substitute these back into the equation: $$144((x + 7)^2 - 49) + 81((y - 4)^2 - 16) = 3312$$ Simplify: $$144(x + 7)^2 - 144 \cdot 49 + 81(y - 4)^2 - 81 \cdot 16 = 3312$$ $$144(x + 7)^2 + 81(y - 4)^2 - 7056 - 1296 = 3312$$ $$144(x + 7)^2 + 81(y - 4)^2 = 3312 + 7056 + 1296$$ $$144(x + 7)^2 + 81(y - 4)^2 = 11664$$ ### Step 4: Divide by 11664 to put in standard form $$\frac{144(x + 7)^2}{11664} + \frac{81(y - 4)^2}{11664} = 1$$ Simplify the fractions: $$\frac{(x + 7)^2}{81} + \frac{(y - 4)^2}{144} = 1$$ This is the standard form of the ellipse equation: $$\frac{(x + 7)^2}{9^2} + \frac{(y - 4)^2}{12^2} = 1$$ ### Answer to Question 5: The standard form is: $$\frac{(x + 7)^2}{9^2} + \frac{(y - 4)^2}{12^2} = 1$$ So, the correct option is **a**. ### Answer to Question 6: The center of the ellipse is at $$(-7, 4)$$, since the equation is in the form $$(x + h)^2/a^2 + (y - k)^2/b^2 = 1$$, where the center is $$(h, k)$$. So, the correct option is **d**. ### Answer to Question 7: The major axis is along the $$y$$-axis because the larger denominator (144) is under the $$y$$-term. The length of the major axis is $$2b = 2 \times 12 = 24$$, so the endpoints are 12 units above and below the center. Thus, the endpoints are at $$(-7, 4 + 12) = (-7, 16)$$ and $$(-7, 4 - 12) = (-7, -8)$$. So, the correct option is **a**. ### Answer to Question 8: The distance between the center and the foci is given by $$c = \sqrt{b^2 - a^2} = \sqrt{144 - 81} = \sqrt{63} \approx 7.94$$. The foci are along the major axis, which is vertical, so the coordinates of the foci are at $$(-7, 4 + 7.94)$$ and $$(-7, 4 - 7.94)$$, approximately $$(-7, 11.94)$$ and $$(-7, -3.94)$$. So, the correct option is **b**. --- Do you want any further details or explanations for these solutions? Here are 5 related questions to consider: 1. How do you find the eccentricity of this ellipse? 2. How does the length of the major axis affect the shape of the ellipse? 3. What would happen if $$a$$ and $$b$$ were equal in an ellipse equation? 4. How can you derive the equation of a hyperbola from an ellipse? 5. Can you graph the equation of this ellipse by hand? **Tip:** To easily spot the center of a conic section like an ellipse, always look for the terms in the form $$(x + h)^2$$ and $$(y - k)^2$$ in the equation.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Ellipse
Standard form
Completing the square
Conic sections
Formulas
(x + h)^2 / a^2 + (y + k)^2 / b^2 = 1 for an ellipse
Distance formula for foci: c = sqrt(b^2 - a^2)
Theorems
Ellipse standard equation theorem
Completing the square method
Suitable Grade Level
Grade 11-12
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