My daughter is stuck

Question

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

Answer 1

Hi Vicky,

 

Rewrite the equation as

x ² -4x +            y² +2y          = 4 

complete the square on the left-hand side using the coefficient of x and y 

The Coeff. of x is  -4, -4/2 = -2   add (-2)² to the left and the right-hand side

The Coeff. of y is  2,  2/2 =  1 add (1)² to the left and the right-hand side

x² -4x +(-2)² + y² + 2y +(1)² = 4 +(-2)² + (1)²

(x - 2)² + (y + 1)² = 4 +4 +1

(x - 2)² + (y + 1)² = 9  compare this to the circle centered at (h,k) with radius r whose equation is 

(x - h)² + (y - k)² = r²

 x ² + y² -4x+2y - 4 =0  is the equation of a circle centered at (2,-1) with radius r = 3

 

 

Answer 2

x² - 4x + y² + 2y = 4

x² - 4x + 4 + y² + 2y + 1 = 9  complete the square

(x - 2)² + (y + 1)² = 9             write a square of binomial

 

Circle centered at (2, -1) with a radius of 3.

Answer 3

The graph of (x-h)²+(y-k)²=r² is a circle with center (h,k) and radius r.  Put the given equation x² - 4x + y² + 2y = 4 in the above form by completing the square.  

 

 (x²-4x+4) + (y²+2y+1) = 4+4+1

 

 (x-2)² + (y+1)² = 9

 

The equation is an equation of the circle centered at (2,-1) with radius 3.