find the center and radius of the circle given the circle x^2+y^2-18x+45=0 is tangent to the line y=(4/3)x-2

Question

find the center and  radius of the circle given the circle x^2+y^2-18x+45=0 is tangent to the line y=(4/3)x-2

anonymous · Answer

complete the square.  $$x^2-18x      +y^2+45 =0$$
 $$x^2-18x +81     +y^2 =-45+81$$
 $$(x-9)^2     +y^2 =-45+9$$
$$(x-9)^2 +(y+0)^2=-36$$
Center: (9,0)
Radius: $$sqrt{36}$$ =6

anonymous · Answer

o I did that and got the same thing but was wondering y they give the tangent line y=4/3x-2  ??  we don't have anything t do with that line??

campbell_st · Answer

seems to be a calculation error 

is you add 81 to complete the square in x, then you need to do it on both sides of the equation.... that's causing the problem with finding the radius

$$(x -9)^2 + y^2 = -45 + 81$$

campbell_st · Answer

so when you have the centre and radius... 

so there are several steps to the next part... 

1. substitute the equation of the line into the circle equation and find how many points of intersection. 

2. find the distance from the centre to the line to see if it is equal to the radius... 

given the way the question is written I'd say option 2. 

so if you have a line in standard form 

$$Ax + By + C = 0$$

the distance from a point $$(x_{0}, y_{0})$$

can be found using 

$$d = \frac{\left| Ax_{0} + By_{0} + C \right|}{\sqrt{A^2 + B^2}}$$

hope it helps...

anonymous · Answer

so$$\frac{ 4 }{ 3 }(9)+1(0)-2/\sqrt{(16/9+1)}=6$$

anonymous · Answer

that equals to 6... which is the radius ...so what does this show ??

anonymous · Answer

@campbell_st

anonymous · Answer

I just want to be clear on this @campbell_st

anonymous · Answer

@amistre64 what does it show... can u say??

amistre64 · Answer

i wouldnt know why the tangent is given ... the equation of a circle is defined already by its center and radius of you complete the squares.

campbell_st · Answer

well if the tangent is y = 4/3x - 2

then 3y = 4x - 6

or 

4x - 3y - 6 = 0

so you need the centre of the circle (9, 0) and the radius 6

so all you are going to do is show that the distance from the tangent to the radius is 6 units

$$d = \frac{\left| 4 \times 9 - 3 \times 0 - 6 \right|}{\sqrt{4^2 + (-3)^2}}$$

so this results in 

$$d = \frac{30}{5} $$

so the distance from the centre to the tangent is 6... so you have the correct equation by completing the square...

campbell_st · Answer

oops should read ... show the distance from the centre to the tangent is 6 units

anonymous · Answer

ok so the distance and the radius is the same... does this always happen..just curious ..i have a test 2morrow

amistre64 · Answer

i do not read in the post that we are to show the distance from the center to the tangent line is required ...

amistre64 · Answer

find radius and center, given _____

otherwise it would read,  find and show that tangent line is yada yada .... at least to me that is

campbell_st · Answer

well its a silly question... 

but if you thing about it, the perpendicular distance from the centre to a tangent will always be a radius

campbell_st · Answer

I meant the initial question is silly

campbell_st · Answer

a tangent has 1 point of contact on the circumference... 

so the distance and radius will be the same.

anonymous · Answer

yea i know i hope no question like this come for this test that i have tomorrow ...

amistre64 · Answer

just make sure you read the test questions, and determine what it is that they are asking for.  once you have answered them, all else is extraneous :)

anonymous · Answer

ok will do that thanks