1. for the given slope-inntercept equation of the line with the following properties parallel to the line y=7; containing the (-3,5)
2. write the equation of the circle in standard form. find the center, radius, intercepts.
x^2+y^2+16x-18y+145=25
please show work
parallel to y =7 is a horizontal line
y = 0x + 5
or y = 5
complete the square a couple of times
x^2 + 16 x = -y^2 + 18 y - 120
x^2 + 16 x + 64 = -y^2 + 18 y - -56
(x+8)^2 + y^2 - 18 y = -56
(x+8)^2 + y^2 - 18 y + 81 = 25
(x+8)^2 + (y-9)^2 = 5^2
center at (-8,9) radius 5 etc
1. To find the equation of the line parallel to y = 7 and passing through the point (-3, 5), we need to determine the slope of the given line.
In the equation y = 7, the coefficient of y is 0, which means the slope of this line is 0 (since the equation is y = mx + b, where m is the slope).
Since parallel lines have the same slope, the line we are looking for also has a slope of 0.
Now we can use the point-slope form of a linear equation: y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope. Plugging in the values we have:
y - 5 = 0(x - (-3))
y - 5 = 0(x + 3)
y - 5 = 0 (since 0 multiplied by any value is 0)
y = 5
Therefore, the equation of the line parallel to y = 7 and passing through (-3, 5), is y = 5.
2. To write the equation of the circle in standard form, we need to complete the square for both the x and y terms.
Given equation: x^2 + y^2 + 16x - 18y + 145 = 25
Rearranging the terms and grouping the x and y terms together, we have:
(x^2 + 16x) + (y^2 - 18y) = 25 - 145
Now, we need to complete the square for both x and y.
For the x terms:
(x^2 + 16x + 64) + (y^2 - 18y) = 25 - 145 + 64
And for the y terms:
(x^2 + 16x + 64) + (y^2 - 18y + 81) = 25 - 145 + 64 + 81
Simplifying, we get:
(x + 8)^2 + (y - 9)^2 = -15 + 225
Now, we have the equation in standard form, (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.
Comparing it to our equation, we can see:
(h, k) = (-8, 9)
r^2 = -15 + 225
r^2 = 210
Taking the square root of both sides, we have:
r ≈ √210
Therefore, the center of the circle is (-8, 9), the radius is approximately √210, and the equation in standard form is (x + 8)^2 + (y - 9)^2 = 210.