Question

Please help!!
Find the center and radius of the given circle and graph it.
x^2+y^2+10y+21=0

I got how to set up the equation but where is a pair of x value for x^2 when there is 10y for y^2?
:(

Answer 1

@phi please help if you are not busy! :)

Answer 2

do you know how to "complete the square" on the y part ?

Answer 3

x^2+y^2+10y=-21
(x^2...??)+(y+2+10y....)=-21
for y, 10/2=5^2=25
(x^2...??)+(y+2+10y+25)=-21+25

Answer 4

I got it till here

Answer 5

I assume you mean (y^2 +10y+25) (not y+2 + 10y +25)

Answer 6

anyway, the point of completing the square is so we can write
(y^2 +10y+25) as (y+5)^2

thus your equation is now
\[ x^2 + (y+5)^2 = 4 \]

Answer 7

next, match that up with the "generic" equation for a circle
(x-a)^2 + (y-b)^2 = r^2

the center of the circle will be at (a,b)
and the radius is r

Answer 8

so the circle would be (0,5)
radius would be 16??

Answer 9

your equation looks a bit like the generic equation , but we have to tweak it to make it look the same form
x^2 + (y+5)^2 = 4
(x-a)^2 + (y-b)^2 = r^2

looking at the right-hand side, we want 4 to match r^2
we know 4 is 2*2 or 2^2. so we would write it this way
x^2 + (y+5)^2 = 2^2
and now 2^2 matches r^2 and we see what r is: it is 2

Answer 10

ohhh I see, then would the circle be correct?

Answer 11

next, we don't have y- b matching with y+5
but we could write y+5 as y - (-5)
that means the same thing, but now we have y - a number
and we can match
x^2 + (y- -5)^2 = 2^2
(x-a)^2 + (y-b)^2 = r^2

and we see b matches up with -5

Answer 12

finally , we can write x as (x-0)
which is still "x", but it now can match up with the generic equation
(x-0)^2 + (y- -5)^2 = 2^2
and we see the center is (0,-5)
and the radius is 2

Answer 13

is it not clear ?

Answer 14

Sorry I was just trying out what you said on my paper. Yes!! I totally get it now

Answer 15

If you don't mind, could you help with this question as well?
Find the slope of the line that is a parallel and perpendicular to the given line.
3x+y=(3x+2)

Answer 16

I know how to find both ways but I'm just confused because the slope is zero.
y=3x-3x+2
y=2?

Answer 17

to do this problem, notice the trick we use
we change
y+5 to y- -5
so that we can match that to y-b

the other trick was to write x as x-0
though if you do enough problems you will remember x^2 by itself means the center of the circle will have 0 for its x value.
and you will remember that y+5 means the y value of the center is -5

if you don't do a lot of these problems, then you have to remember the tricks.

Answer 18

3x+y=(3x+2)
yes that simplifies to
y=2

y=2 means points like (0,2), (1,2), (2,2) all lie on that line
if you plot those points you see a line going horizontal
(or as you noted, its slope is 0, and that means a line going sideways)

they want a line parallel to y=2
pick slope=0 but use a different y intercept.

Answer 19

they also want a perpendicular line to y=2
that means you want a line that forms a right angle i.e. 90 degrees
that would be a straight up/down vertical line

do you know the equation of a vertical line ? you write x= 0 or x=1 or x=2
all of those are vertical lines

Answer 20

the slope of a vertical line is called "undefined"

reading the question, they want the slope of a parallel and perpendicular line
the answer is : 0 and "undefined"

Answer 21

Thanks!