Write 2x^2+2y^2-16x-8y+8=0 in the standard form

Question

write the equation in standard form

Answer 1

2x^2+2y^2-16x-8y+8 = 0

Divide to each side by 2.

(2x^2+2y^2-16x-8y+8)/2  = 0/2

x^2+y^2-8x-4y+4 = 0

Compare it to circle equation form x^2+y^2+2gx+2fy+c = 0

Here centere is (-g,-f)

r = sqrt (g^2+f^2-c)

Given circle centere is (4,2) and radious is 4.

Standard form is x^2+y^2-8x-4y+4 = 0.

 

Comments

x ² +y ²+ 2gx + 2fy + c = 0 is general equation of a circle.

Standard form of a circle equation is .

Answer 2

The equation is 2x ² + 2y ² - 16x - 8y + 8 = 0

Here x ² coefficient is 2 , for perfect square make x ² and y ²coefficient 1 by dividing each side by 2.

x ² + y ² - 8x - 4y  = -4

To change the expression into a perfect square  add (half the x coefficient)² and add (half the y coefficient)²to each side of the expression.

Here x coefficient = -8. so, (half the x coefficient)² = (-8/2)²= 16

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

Add 16 and 4 to each side

x ² - 8x +16 + y ²  - 4y + 4 = - 4 +16 + 4

(x - 4)² + (y - 2)² = 16

(x - 4)² + (y - 2)² = 4²

Compare it to standard form of a circle equation is

Where center is (h ,k ) and radius of circle is r .

Radius of circle r = 4 and center of the circle is (4, 2).

Standard form of 2x ² + 2y ² - 16x - 8y + 8 = 0 is (x - 4)² + (y - 2)² = 4² .