A circle has the equation x2 + y2 – x + 10y + 13 = 0. What is the radius of this circle?

Answered by 560047712

The radius would be 4. Move all variables to the left side and all constants to the right side.x2 +y2 −4x−10y=−13+0 x2 +y2 -4x-10y=-13+0 Add −13 -13  and 00  to get −13 -13 .x2 +y2 −4x−10y=−13 x2 +y2 -4x-10y=-13 Complete the square for x2 −4x x2 -4x .Tap for more steps...(x−2) 2 −4 (x-2) 2 -4 Substitute (x−2) 2 −4 (x-2) 2 -4  for x2 −4x x2 -4x  in the equation x2 +y2 −4x−10y=−13 x2 +y2 -4x-10y=-13 .(x−2) 2 −4+y2 −10y=−13 (x-2) 2 -4+y2 -10y=-13 Move −4 -4  to the right side of the equation by adding 44  to both sides.(x−2) 2 +y2 −10y=−13+4 (x-2) 2 +y2 -10y=-13+4 Complete the square for y2 −10y y2 -10y .Tap for more steps...(y−5) 2 −25 (y-5) 2 -25 Substitute (y−5) 2 −25 (y-5) 2 -25  for y2 −10y y2 -10y  in the equation x2 +y2 −4x−10y=−13 x2 +y2 -4x-10y=-13 .(x−2) 2 +(y−5) 2 −25=−13+4 (x-2) 2 +(y-5) 2 -25=-13+4 Move −25 -25  to the right side of the equation by adding 2525  to both sides.(x−2) 2 (y−5) 2 =−13+4+25 (x-2) 2 (y-5) 2 =-13+4+25 Add −13 -13  and 44  to get −9 -9 .(x−2) 2 +(y−5) 2 =−9+25 (x-2) 2 +(y-5) 2 =-9+25 Add −9 -9  and 2525  to get 1616 .(x−2) 2 +(y−5) 2 =16 (x-2) 2 +(y-5) 2 =16 This is the form of a circle. Use this form to determine the center and radius of the circle.(x−h) 2 +(y−k) 2 =r2 r=4 r=4 h=2 h=2 k=5

Answered by altavistard

Hint:  Make your "x2 + y2 – x + 10y + 13 = 0" clearer by using " ^ " to indicate exponentiation: x^2 + y^2 – x + 10y + 13 = 0 Rearrange this so that x terms are together and y terms are also (separately): x^2  - 1x + y^2 + 10y + 13 = 0 Complete the square twice:  once for x and once for y: (x^2 - 1x + (-1/2)^2 - (-1/2)^2 + y^2 + 10y + 25 - 25 = - 13   Condense:     (x-1/2)^2 + (y+5)^2  - 1/4 - 25 = -13 Combine -1/4, -25 and -13:    (x-1/2)^2 + (y+5)^2  - 1/4 - 25 = -13 + 1/4 + 25  =  11 3/4 = 47/4 Then we have    (x-1/2)^2 + (y+5)^2  =  47/4, and r^2 = 47/4.  Then the radius of the circle is sqrt(47/4) = sqrt(47)/2  (answer)