The equation of the ellipse with its centre at (1, 2), focus at (6, 2) and passing through the point (4 ,6) is `(x-1)^(2)/a^(2)+(y-2)^(2)/b^(2)=1,` then

Options

A. `a^(2)=1,b^(2)=25`
B. `a^(2)=25,b^(2)=20`
C. `a^(2)=20,b^(2)=25`
D. None of these

Answer

To find the equation of the ellipse with its center at (1, 2), focus at (6, 2), and passing through the point (4, 6), we can follow these steps:

### Step 1: Identify the parameters of the ellipse
The center of the ellipse is given as (H, K) = (1, 2). The focus is at (6, 2). Since the focus lies on the horizontal line with the center, we can conclude that the major axis is horizontal.

### Step 2: Calculate the distance from the center to the focus
The distance \( c \) from the center to the focus can be calculated as:
\[ c = |6 - 1| = 5 \]

### Step 3: Use the relationship between a, b, and c
For an ellipse, the relationship between the semi-major axis \( a \), semi-minor axis \( b \), and the distance to the focus \( c \) is given by:
\[ c^2 = a^2 - b^2 \]
Substituting the value of \( c \):
\[ 5^2 = a^2 - b^2 \implies 25 = a^2 - b^2 \tag{1} \]

### Step 4: Write the general equation of the ellipse
The general equation of the ellipse centered at (H, K) is:
\[ \frac{(x - H)^2}{a^2} + \frac{(y - K)^2}{b^2} = 1 \]
Substituting H and K:
\[ \frac{(x - 1)^2}{a^2} + \frac{(y - 2)^2}{b^2} = 1 \tag{2} \]

### Step 5: Substitute the point (4, 6) into the ellipse equation
Since the ellipse passes through the point (4, 6), we substitute \( x = 4 \) and \( y = 6 \) into equation (2):
\[ \frac{(4 - 1)^2}{a^2} + \frac{(6 - 2)^2}{b^2} = 1 \]
Calculating the squares:
\[ \frac{3^2}{a^2} + \frac{4^2}{b^2} = 1 \implies \frac{9}{a^2} + \frac{16}{b^2} = 1 \tag{3} \]

### Step 6: Solve the system of equations
Now we have two equations:
1. \( a^2 - b^2 = 25 \) (from step 3)
2. \( \frac{9}{a^2} + \frac{16}{b^2} = 1 \) (from step 5)

From equation (1), we can express \( a^2 \) in terms of \( b^2 \):
\[ a^2 = 25 + b^2 \]
Substituting this into equation (3):
\[ \frac{9}{25 + b^2} + \frac{16}{b^2} = 1 \]

### Step 7: Clear the fractions
Multiply through by \( b^2(25 + b^2) \):
\[ 9b^2 + 16(25 + b^2) = b^2(25 + b^2) \]
Expanding:
\[ 9b^2 + 400 + 16b^2 = 25b^2 + b^4 \]
Combine like terms:
\[ 25b^2 + 400 = 25b^2 + b^4 \]
This simplifies to:
\[ b^4 - 400 = 0 \implies b^4 = 400 \implies b^2 = 20 \]

### Step 8: Find \( a^2 \)
Using \( b^2 = 20 \) in equation (1):
\[ a^2 = 25 + 20 = 45 \]

### Step 9: Write the final equation of the ellipse
Now we can write the equation of the ellipse:
\[ \frac{(x - 1)^2}{45} + \frac{(y - 2)^2}{20} = 1 \]

### Final Answer
The equation of the ellipse is:
\[ \frac{(x - 1)^2}{45} + \frac{(y - 2)^2}{20} = 1 \]