Find the center and foci of an ellipse with the equation

Question

9x^2+16y^2-18x+64y=71.

Answer 1

The equation of ellipse is .

The standard form of equation of ellipse is ,

where (h, k) = center, foci = (±c, 0), major axis length = 2a, minor axis length = 2b and the relation between a, b and c is .

Write the above equation in standard form.

 To change the expressions into a perfect square trinomial add (half the x coefficient)² to each side of the expression

 Here x coefficient = 2. so, (half the x coefficient)² = (2/2)²= 1

and y coefficient = 4. so, (half the y coefficient)² = (4/2)²= 4.

Add 9(1) and 16(4) = 64 to each side.

To find the vlue of c, substitute the values of a = 4 and b = 3 in the equation .

.

Center (h, k) = (1, - 2) and Foci : .

Comments

Equations of Ellipses with Centers at (h, k).

Standard Form of Equation	(x - h)2/a2 + (y - k)2/b2 = 1	(y - k)2/a2 + (x - h)2/b2 = 1
Direction of Major Axis	horizontal	vertical
Length of Major Axis	2a units	2a units
Length of Minor Axis	2b units	2b units
Foci	(h ± c, k)	(h, k ± c)
The relation between a, b and c		c2 = a2 - b2

The standard form of equation of ellipse 9x² + 16y² - 18x + 64y = 71 is (x - 1)²/4² + (y + 2)²/3² = 1.

Center = (h, k) = (1, - 2) and c = ± √7.

Foci = (h ± c, k) = (1 ± √7, - 2) = (1 + √7, - 2) and (1 - √7, - 2).