# How do you find the coordinates of the center, foci, the length of the major and minor axis given 7x^2+3y^2-28x-12y=-19?

May 29, 2018

Center is at $\left(2 , 2\right)$, Major axis length is $2 \sqrt{7}$ , Minor axis length is $2 \sqrt{3}$, Focii are at $\left(2 , 4\right) \mathmr{and} \left(2 , 0\right)$ .

#### Explanation:

$7 {x}^{2} + 3 {y}^{2} - 28 x - 12 y = - 19$ or

$7 \left({x}^{2} - 4 x\right) + 3 \left({y}^{2} - 4 y\right) = - 19$ or

$7 \left({x}^{2} - 4 x + 4\right) + 3 \left({y}^{2} - 4 y + 4\right) = 28 + 12 - 19$ or

$7 {\left(x - 2\right)}^{2} + 3 {\left(y - 2\right)}^{2} = 21$ or

$\frac{7 {\left(x - 2\right)}^{2}}{21} + \frac{3 {\left(y - 2\right)}^{2}}{21} = \frac{21}{21}$ or

(x-2)^2/3+( y-2)^2/7 =1 ; 7 >3 This is standard equation of

vertical ellipse with center at $\left({x}_{1} = 2 , {y}_{1} = 2\right)$and equation is

 (x-x_1)^2/b^2+(y-y_1)^2/a^2=1; b = sqrt 3, a= sqrt 7

Major axis length is $2 a = 2 \sqrt{7}$ , Minor axis length is

2 b = 2 sqrt3 ; c^2= a^2-b^2= 7- 3 = 4 :. c= +- 2

Focii are at $2 , \left(2 + 2\right) \mathmr{and} 2 , \left(2 - 2\right) \mathmr{and} \left(2 , 4\right) \mathmr{and} \left(2 , 0\right)$ on

major axis .

graph{7 x^2 +3 y^2 -28 x -12 y= -19 [-10, 10, -5, 5]} [Ans]