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Sagot :
To find the standard form of the equation of the ellipse with the given foci and vertices, we need to follow a series of steps systematically. Let's break it down:
### Step 1: Identify Key Values
- Distance between the foci: The given foci are [tex]\( (-6, 0) \)[/tex] and [tex]\( (6, 0) \)[/tex]. The distance between these points is given by:
[tex]\[ \text{Distance between foci} = 2c = 6 - (-6) = 12 \][/tex]
Therefore, the distance [tex]\( c \)[/tex] is:
[tex]\[ c = \frac{12}{2} = 6 \][/tex]
- Distance between the vertices: The given vertices are [tex]\( (-9, 0) \)[/tex] and [tex]\( (9, 0) \)[/tex]. The total distance between these points is:
[tex]\[ \text{Distance between vertices} = 2a = 9 - (-9) = 18 \][/tex]
Therefore, the distance [tex]\( a \)[/tex] is:
[tex]\[ a = \frac{18}{2} = 9 \][/tex]
### Step 2: Use the Relationship in Ellipses
For an ellipse, the relationship between [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] is given by
[tex]\[ c^2 = a^2 - b^2 \][/tex]
Here, we already know [tex]\( a = 9 \)[/tex] and [tex]\( c = 6 \)[/tex]. We can use these values to find [tex]\( b \)[/tex].
### Step 3: Calculate [tex]\( b \)[/tex]
First, compute [tex]\( a^2 \)[/tex]:
[tex]\[ a^2 = 9^2 = 81 \][/tex]
Next, compute [tex]\( c^2 \)[/tex]:
[tex]\[ c^2 = 6^2 = 36 \][/tex]
Now, solve for [tex]\( b^2 \)[/tex]:
[tex]\[ b^2 = a^2 - c^2 = 81 - 36 = 45 \][/tex]
Finally, take the square root to find [tex]\( b \)[/tex]:
[tex]\[ b = \sqrt{45} \approx 6.708 \][/tex]
### Step 4: Write the Equation in Standard Form
The standard form of an ellipse equation with its center at [tex]\( (h, k) \)[/tex], major axis along the x-axis, and lengths [tex]\( a \)[/tex] and [tex]\( b \)[/tex] is:
[tex]\[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \][/tex]
Given the center is at [tex]\( (0, 0) \)[/tex], [tex]\( h = 0 \)[/tex] and [tex]\( k = 0 \)[/tex]. Substitute [tex]\( a^2 \)[/tex] and [tex]\( b^2 \)[/tex] into the equation:
[tex]\[ \frac{(x - 0)^2}{81} + \frac{(y - 0)^2}{45} = 1 \][/tex]
Therefore, the standard form of the equation of the ellipse is:
[tex]\[ \left( \frac{x^2}{81} \right) + \left( \frac{y^2}{45} \right) = 1 \][/tex]
This is your final equation for the ellipse given the specified foci and vertices.
### Step 1: Identify Key Values
- Distance between the foci: The given foci are [tex]\( (-6, 0) \)[/tex] and [tex]\( (6, 0) \)[/tex]. The distance between these points is given by:
[tex]\[ \text{Distance between foci} = 2c = 6 - (-6) = 12 \][/tex]
Therefore, the distance [tex]\( c \)[/tex] is:
[tex]\[ c = \frac{12}{2} = 6 \][/tex]
- Distance between the vertices: The given vertices are [tex]\( (-9, 0) \)[/tex] and [tex]\( (9, 0) \)[/tex]. The total distance between these points is:
[tex]\[ \text{Distance between vertices} = 2a = 9 - (-9) = 18 \][/tex]
Therefore, the distance [tex]\( a \)[/tex] is:
[tex]\[ a = \frac{18}{2} = 9 \][/tex]
### Step 2: Use the Relationship in Ellipses
For an ellipse, the relationship between [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] is given by
[tex]\[ c^2 = a^2 - b^2 \][/tex]
Here, we already know [tex]\( a = 9 \)[/tex] and [tex]\( c = 6 \)[/tex]. We can use these values to find [tex]\( b \)[/tex].
### Step 3: Calculate [tex]\( b \)[/tex]
First, compute [tex]\( a^2 \)[/tex]:
[tex]\[ a^2 = 9^2 = 81 \][/tex]
Next, compute [tex]\( c^2 \)[/tex]:
[tex]\[ c^2 = 6^2 = 36 \][/tex]
Now, solve for [tex]\( b^2 \)[/tex]:
[tex]\[ b^2 = a^2 - c^2 = 81 - 36 = 45 \][/tex]
Finally, take the square root to find [tex]\( b \)[/tex]:
[tex]\[ b = \sqrt{45} \approx 6.708 \][/tex]
### Step 4: Write the Equation in Standard Form
The standard form of an ellipse equation with its center at [tex]\( (h, k) \)[/tex], major axis along the x-axis, and lengths [tex]\( a \)[/tex] and [tex]\( b \)[/tex] is:
[tex]\[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \][/tex]
Given the center is at [tex]\( (0, 0) \)[/tex], [tex]\( h = 0 \)[/tex] and [tex]\( k = 0 \)[/tex]. Substitute [tex]\( a^2 \)[/tex] and [tex]\( b^2 \)[/tex] into the equation:
[tex]\[ \frac{(x - 0)^2}{81} + \frac{(y - 0)^2}{45} = 1 \][/tex]
Therefore, the standard form of the equation of the ellipse is:
[tex]\[ \left( \frac{x^2}{81} \right) + \left( \frac{y^2}{45} \right) = 1 \][/tex]
This is your final equation for the ellipse given the specified foci and vertices.
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