Find the center and radius of the circle with equation x2 y

Find the center and radius of the circle with equation: x^2 + y^2 + 6x - 10y + 18 = 0. For the curve with equation y = x^2 - 5: Find the x-intercepts (list both coordinates for each one). Find the y-intercept (list both coordinates). For the curve with equation y^2 = x - 6, check for symmetries. State Yes or No for each of the following symmetries, and briefly justify each answer. Symmetry with respect to the x-axis:______(Yes or No; justify below)

Solution

10)   x^2 + y^2 + 6x - 10y + 18 = 0

standard equation of circle is

(x-h)^2 + (y-k)^2 = r^2

where h,k are centre

r = radius of the circle

writing the given equation in standrad form

x^2 + y^2 + 6x - 10y + 18 = 0

( x+ 3)^2 - 9 + (y-5)^2 - 25 + 18 = 0

(x+3)^2 + (y-5)^2 = 16

centre = ( -3 , 5)

radius = 4

11)   y = x^2 - 5

a) x intercept ( plug y = 0 )

0 = x^2 - 5

x^2 = 5

x = +sqrt 5

x = - sqrt 5

x intercepts are ( sqrt 5 , 0) and ( - sqrt 5 , 0 )

b) y intercept ( plug x =0 )

y = 0 - 5

y intercept = ( 0,-5)

12) y^2 = x-6

y = sqrt ( x-6 )

function is symmetric with respect to x axis if f(x) = -f(x)

therefore, this function is not symmetric with respect to x axis