Question

Find the equation of the line that is/are parallel to the line 3x-y=4 and tangent to the circle x^2+y^2=9

Answer 1

@amistre64

Answer 2

this was one of my test question that i had to do today.. and i think i did it wrong.. so i want to see how exactly this question is solved...

Answer 3

what was your approach?

Answer 4

well i rewrite 3x-y=4 to -y=-3x+4 to y=3x-4 so the slope =3
then i just find guess some points that satisfy the equation of the circle like (3,0) and (0,3) then used the slope with those points to write the equation because center (0,0) and (3,0)
give 9 so.. i did y-3=3(x-0)
y=3x+3 and y-0=3(x-3).. so y=3x+9

Answer 5

that wat i did since i was running out of time

Answer 6

what is the derivative of the circle equation?

Answer 7

or, since the circle is centered about the origin, perp the slope

y = -x/3 and place that into the circle to solve for points of tangency is one approach

Answer 8

2x+2xdy/dx=0
dy/dx=-2x/2
dy/dy=-x

Answer 9

2x + 2y y' = 0

y' = -x/y

Answer 10

so wats the equation...none of the equation i wrote was correct?

Answer 11

i like my circle approach better tho ...

Answer 12

but it says that the line 3x-y=4 is tangent to the circle

Answer 13

no, it doesnt ... its says the tangent lines are parallel to that line

slope of 3 is fine

but derivative of the circle equation is y' = -x/y

y = sqrt(9-x^2)

Answer 14

\[-\frac{x}{\sqrt{9-x^2}}=3\]
\[-x=3\sqrt{9-x^2}\]
\[x^2=9(9-x^2)\]
what is x?

Answer 15

10x^2 = 81

x^2 = 81/10; x=+- 9sqrt(.1)

Answer 16

81-9x^2-x^2=0
-10x^2=-81
x^2=81/10
x=+or minus 9/sq.rt 10

Answer 17

then plugging that in we can determine y

Answer 18

x^2 + y^2 = 9

let y=-1/3x

x^2 + (-1/3x)^2 = 9

x^2 + 1/9 x^2 = 9

9x^2 + x^2 = 81 .. same results

Answer 19

o woo so y=-1/3x would be the equation

Answer 20

well, thats just the approach that made it simpler for me

the line given has a slope of 3, a line thru the center of the circle, with a perp slope, will pass thru the circle at the points of tangency ...

since the circle is centered at the origin, let y = -1/3 x and see where it hits the circle ...

Answer 21

ok

Answer 22

thanks for clearing this up for me

Answer 23

yep