Thank you for visiting 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. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
* Rewrite the ellipse equation in standard form: $\frac{x^2}{625} + \frac{y^2}{49} = 1$.
* Identify $a$ and $b$: $a = 25$, $b = 7$.
* Calculate the major axis length $2a = 50$, the minor axis length $2b = 14$, and $c = \sqrt{a^2 - b^2} = 24$.
* State the foci coordinates: $(\pm 24, 0)$.
The length of the major axis is $\boxed{50}$ units. The length of the minor axis is $\boxed{14}$ units. The foci are located at $(\pm \boxed{24}, \boxed{0})$.
### Explanation
1. Analyze the given equation
We are given the equation of an ellipse: $49x^2 + 625y^2 = 30625$. Our goal is to find the length of the major axis, the length of the minor axis, and the coordinates of the foci.
2. Rewrite in standard form
First, we need to rewrite the equation in standard form. To do this, we divide both sides of the equation by 30625:
$$\frac{49x^2}{30625} + \frac{625y^2}{30625} = 1$$
Simplifying, we get:
$$\frac{x^2}{625} + \frac{y^2}{49} = 1$$
3. Find a and b
Now we can identify $a^2$ and $b^2$. We have $a^2 = 625$ and $b^2 = 49$. Taking the square root of both, we get $a = \sqrt{625} = 25$ and $b = \sqrt{49} = 7$.
4. Calculate major and minor axis lengths
The length of the major axis is $2a = 2(25) = 50$ units. The length of the minor axis is $2b = 2(7) = 14$ units.
5. Calculate the focal distance
To find the foci, we need to calculate $c$, the distance from the center to each focus. We use the formula $c^2 = a^2 - b^2$. In this case, $c^2 = 625 - 49 = 576$. Taking the square root, we get $c = \sqrt{576} = 24$.
6. Determine the foci coordinates
Since the major axis is along the x-axis, the foci are located at $(\pm c, 0)$, which is $(\pm 24, 0)$.
7. State the final answer
Therefore, the length of the major axis is 50 units, the length of the minor axis is 14 units, and the foci are located at $(\pm 24, 0)$.
### Examples
Ellipses are commonly used in architecture and engineering, such as in the design of bridges and arches. Understanding the properties of an ellipse, like the lengths of its major and minor axes and the location of its foci, is crucial for ensuring structural stability and aesthetic appeal. For example, architects use elliptical arches to distribute weight evenly, and engineers use elliptical shapes in bridge designs to withstand stress and wind.
* Identify $a$ and $b$: $a = 25$, $b = 7$.
* Calculate the major axis length $2a = 50$, the minor axis length $2b = 14$, and $c = \sqrt{a^2 - b^2} = 24$.
* State the foci coordinates: $(\pm 24, 0)$.
The length of the major axis is $\boxed{50}$ units. The length of the minor axis is $\boxed{14}$ units. The foci are located at $(\pm \boxed{24}, \boxed{0})$.
### Explanation
1. Analyze the given equation
We are given the equation of an ellipse: $49x^2 + 625y^2 = 30625$. Our goal is to find the length of the major axis, the length of the minor axis, and the coordinates of the foci.
2. Rewrite in standard form
First, we need to rewrite the equation in standard form. To do this, we divide both sides of the equation by 30625:
$$\frac{49x^2}{30625} + \frac{625y^2}{30625} = 1$$
Simplifying, we get:
$$\frac{x^2}{625} + \frac{y^2}{49} = 1$$
3. Find a and b
Now we can identify $a^2$ and $b^2$. We have $a^2 = 625$ and $b^2 = 49$. Taking the square root of both, we get $a = \sqrt{625} = 25$ and $b = \sqrt{49} = 7$.
4. Calculate major and minor axis lengths
The length of the major axis is $2a = 2(25) = 50$ units. The length of the minor axis is $2b = 2(7) = 14$ units.
5. Calculate the focal distance
To find the foci, we need to calculate $c$, the distance from the center to each focus. We use the formula $c^2 = a^2 - b^2$. In this case, $c^2 = 625 - 49 = 576$. Taking the square root, we get $c = \sqrt{576} = 24$.
6. Determine the foci coordinates
Since the major axis is along the x-axis, the foci are located at $(\pm c, 0)$, which is $(\pm 24, 0)$.
7. State the final answer
Therefore, the length of the major axis is 50 units, the length of the minor axis is 14 units, and the foci are located at $(\pm 24, 0)$.
### Examples
Ellipses are commonly used in architecture and engineering, such as in the design of bridges and arches. Understanding the properties of an ellipse, like the lengths of its major and minor axes and the location of its foci, is crucial for ensuring structural stability and aesthetic appeal. For example, architects use elliptical arches to distribute weight evenly, and engineers use elliptical shapes in bridge designs to withstand stress and wind.
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