Question

determine the radius and coordinates of the centre of circles
x^2 +2x+y^2 _8y=_8​

Answer 1

Given Equation,

$\sf \: {x}^{2} + 2x + {y}^{2} - 8y = - 8$

We have to convert the expression into the form,

$\sf \: {(x - a)}^{2} + {(y - b)}^{2} = {r}^{2} - - - - - - (1)$

Here,

• (a,b) are the coordinates of centre
• r is the radius of circle

Adding and Subtracting 1 and 16 to the given equation,

$\dashrightarrow \sf \:( {x}^{2} + 2x +1) - 1 + ({y}^{2} - 8y + 16) - 16 = - 8 \\ \\ \dashrightarrow \sf \: (x + 1) {}^{2} + (y - 4) {}^{2} - 17 = - 8 \\ \\ \dashrightarrow \sf \: (x + 1) {}^{2} + (y - 4) {}^{2} = 17 - 8 \\ \\ \dashrightarrow \sf \: (x + 1) {}^{2} + (y - 4) {}^{2} = 9 \\ \\ \dashrightarrow \sf \: (x + 1) {}^{2} + (y - 4) {}^{2} = {3}^{2} - - - - - - - (2)$

Comparing (1) and (2),

• radius = 3 units
• coordinates of centre = (-1,4)