Find the equation of the ellipse from the following data: axis is coincident with x = 1, centre (1, 5), focus is (1, 8) and the sum of the focal distances of a point on the ellipse is 12.]

Answer

To find the equation of the ellipse given the data, we can follow these steps:

### Step 1: Understand the given data
- Center of the ellipse: \( (1, 5) \)
- Focus of the ellipse: \( (1, 8) \)
- The sum of the distances from any point on the ellipse to the foci is \( 12 \).

### Step 2: Identify the orientation of the ellipse
Since the center and the focus share the same x-coordinate, the major axis of the ellipse is vertical. Thus, the standard form of the equation of the ellipse will be:
\[ \frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1 \]
where \( (h, k) \) is the center, \( a \) is the semi-major axis, and \( b \) is the semi-minor axis.

### Step 3: Calculate the distance from the center to the focus
The distance \( c \) from the center to the focus is given by:
\[ c = \sqrt{(x_f - x_c)^2 + (y_f - y_c)^2} \]
Substituting the coordinates of the center \( (1, 5) \) and focus \( (1, 8) \):
\[ c = \sqrt{(1 - 1)^2 + (8 - 5)^2} = \sqrt{0 + 3^2} = 3 \]
Thus, \( c = 3 \).

### Step 4: Relate the semi-major axis \( a \), semi-minor axis \( b \), and \( c \)
For ellipses, the relationship between the semi-major axis \( a \), semi-minor axis \( b \), and the distance to the focus \( c \) is given by:
\[ c^2 = a^2 - b^2 \]
We know \( c = 3 \), so:
\[ c^2 = 9 \]
Thus, we have:
\[ 9 = a^2 - b^2 \quad \text{(1)} \]

### Step 5: Use the sum of the focal distances
The sum of the distances from any point on the ellipse to the foci is \( 2a \). Given that this sum is \( 12 \):
\[ 2a = 12 \implies a = 6 \]

### Step 6: Substitute \( a \) into the equation (1)
Now substituting \( a = 6 \) into the equation (1):
\[ 9 = 6^2 - b^2 \]
\[ 9 = 36 - b^2 \]
\[ b^2 = 36 - 9 = 27 \]

### Step 7: Write the equation of the ellipse
Now we have \( a^2 = 36 \) and \( b^2 = 27 \). The center \( (h, k) = (1, 5) \). Therefore, the equation of the ellipse is:
\[ \frac{(x - 1)^2}{27} + \frac{(y - 5)^2}{36} = 1 \]

### Final Equation
Thus, the final equation of the ellipse is:
\[ \frac{(x - 1)^2}{27} + \frac{(y - 5)^2}{36} = 1 \]

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