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Answer :
To find the key features of the ellipse given by the equation [tex]\(49x^2 + 625y^2 = 30625\)[/tex], we need to rewrite it in the standard form of an ellipse equation, which is [tex]\(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)[/tex].
1. Rewrite the Equation in Standard Form:
Divide the entire equation by 30625 to set it equal to 1:
[tex]\[
\frac{49x^2}{30625} + \frac{625y^2}{30625} = 1
\][/tex]
Simplifying the denominators, we have:
[tex]\[
\frac{x^2}{625} + \frac{y^2}{49} = 1
\][/tex]
2. Identify [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex]:
From the equation [tex]\(\frac{x^2}{625} + \frac{y^2}{49} = 1\)[/tex], we have:
- [tex]\(a^2 = 625\)[/tex]
- [tex]\(b^2 = 49\)[/tex]
3. Calculate [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- [tex]\(a = \sqrt{625} = 25\)[/tex]
- [tex]\(b = \sqrt{49} = 7\)[/tex]
4. Determine the Lengths of the Major and Minor Axes:
Since [tex]\(a > b\)[/tex], the major axis is along the x-axis.
- Length of the major axis = [tex]\(2a = 2 \times 25 = 50\)[/tex] units
- Length of the minor axis = [tex]\(2b = 2 \times 7 = 14\)[/tex] units
5. Calculate the Foci Locations:
The distance [tex]\(c\)[/tex] from the center to each focus is calculated using [tex]\(c^2 = a^2 - b^2\)[/tex].
- [tex]\(c^2 = 625 - 49 = 576\)[/tex]
- [tex]\(c = \sqrt{576} = 24\)[/tex]
Since the major axis is along the x-axis, the foci are located at [tex]\((\pm c, 0)\)[/tex].
- Foci: [tex]\((\pm 24, 0)\)[/tex]
In conclusion, the complete answers are:
- The length of the major axis is 50 units.
- The length of the minor axis is 14 units.
- The foci are located at [tex]\((\pm 24, 0)\)[/tex].
1. Rewrite the Equation in Standard Form:
Divide the entire equation by 30625 to set it equal to 1:
[tex]\[
\frac{49x^2}{30625} + \frac{625y^2}{30625} = 1
\][/tex]
Simplifying the denominators, we have:
[tex]\[
\frac{x^2}{625} + \frac{y^2}{49} = 1
\][/tex]
2. Identify [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex]:
From the equation [tex]\(\frac{x^2}{625} + \frac{y^2}{49} = 1\)[/tex], we have:
- [tex]\(a^2 = 625\)[/tex]
- [tex]\(b^2 = 49\)[/tex]
3. Calculate [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- [tex]\(a = \sqrt{625} = 25\)[/tex]
- [tex]\(b = \sqrt{49} = 7\)[/tex]
4. Determine the Lengths of the Major and Minor Axes:
Since [tex]\(a > b\)[/tex], the major axis is along the x-axis.
- Length of the major axis = [tex]\(2a = 2 \times 25 = 50\)[/tex] units
- Length of the minor axis = [tex]\(2b = 2 \times 7 = 14\)[/tex] units
5. Calculate the Foci Locations:
The distance [tex]\(c\)[/tex] from the center to each focus is calculated using [tex]\(c^2 = a^2 - b^2\)[/tex].
- [tex]\(c^2 = 625 - 49 = 576\)[/tex]
- [tex]\(c = \sqrt{576} = 24\)[/tex]
Since the major axis is along the x-axis, the foci are located at [tex]\((\pm c, 0)\)[/tex].
- Foci: [tex]\((\pm 24, 0)\)[/tex]
In conclusion, the complete answers are:
- The length of the major axis is 50 units.
- The length of the minor axis is 14 units.
- The foci are located at [tex]\((\pm 24, 0)\)[/tex].
Thank you for reading the article An ellipse has the equation tex 49x 2 625y 2 30 625 tex Complete the statements to describe key features of the ellipse 1 The. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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