Let the equation of the required ellipse be

…(i)

Given:

Coordinates of foci = (±3, 0) …(ii)

We know that,

Coordinates of foci = (±c, 0) …(iii)

∴ From eq. (ii) and (iii), we get

c = 3

We know that,

c² = a² – b²

⇒ (3)² = a² – b²

⇒ 9 = a² – b²

⇒ b² = a² – 9 …(iv)

Given that ellipse passing through the points (4, 1)

So, point (4, 1) will satisfy the eq. (i)

Taking point (4, 1) where x = 4 and y = 1

Putting the values in eq. (i), we get

[from (iv)]

⇒ 16a² – 144 + a² = a²(a² – 9)

⇒ 17a² – 144 = a⁴ – 9a²

⇒ a⁴ – 9a² – 17a² + 144 = 0

⇒ a⁴ – 26a² + 144 = 0

⇒ a⁴ – 8a² – 18a² + 144 = 0

⇒ a²(a² – 8) – 18(a² – 8) = 0

⇒ (a² – 8)(a² – 18) = 0

⇒ a² – 8 = 0 or a² – 18 = 0

⇒ a² = 8 or a² = 18

If a² = 8 then

b² = 8– 9

= - 1

Since the square of a real number cannot be negative. So, this is not possible

If a² = 18 then

b² = 18 – 9

= 9

So, equation of ellipse if a² = 18 and b² = 9