The equation of the ellipse with its centre at `(1, 2)`, one focus at `(6, 2)` and passing through the point `(4, 6)` is-

Options

A. `((x - 1)^(2))/(45) + ((y - 2)^(2))/(20) = 1`
B. `((x - 1)^(2))/(20) + ((y - 2)^(2))/(45) = 1`
C. `((x + 1)^(2))/(45) + ((y + 2)^(2))/(20) = 1`
D. none of these

Answer

A

Explanation

Let the equation of the ellipse be
`((x - 1)^(2))/(a^(2)) + ((y - 2)^(2))/(b^(2)) = 1" "…(i)`
It passes through (4, 6).
`therefore 9/a^(2) + 16/b^(2) = 1" "` …(ii)
Let e be the eccentricity of the ellipse. Then,
ae = Distance between (1, 2) and (6, 2)
`rArr ae = 5`
`rArr a^(2)e^(2) = 25 rArr a^(2) - b^(2) = 25 rArr a^(2) = 25 + b^(2)" "...(iii)`
Solving (ii) and (iii), we get `a^(2) = 45, and b^(2) = 20`
Hence, the equation of the ellipse is `((x - 1)^(2))/(45) + ((y - 2)^(2))/(20) = 1`