Question

Find equation of ellipse given foci (3 0) and passng through (4,1)

Answer 1

Let equation of ellipse is in the form of x²/a² + y²/b² = 1 ,
then foci ( ± c , 0) where c = a² - b²

here Given , foci ( ±3 , 0) = ( ± c , 0)
so, c = 3
e.g., a² - b² = c² = 3²
a² - b² = 9 ------------(1)

again, ellipse passing through ( 4, 1) . it means (4, 1) will satisfy equation of ellipse .

4²/a² + 1²/b² = 1
16/a² + 1/b² = 1
16b² + a² = a²b² -----------(2)

from equation (1) and equation (2)

16b² + b² + 9 = (b² + 9)b²
17b² + 9 = b⁴ + 9b²
b⁴ - 8b² - 9 = 0
b⁴ - 9b² + b² - 9 =0
b²(b² - 9) + (b² - 9) = 0
(b² - 9)(b² + 1) = 0
b² = 9 , b² ≠ -1
b = 3 because b can't be negative .
a² = b² + 9 = 18
hence,
x²/18 + y²/9 = 1