circle problem

[klreed2]

Let C be the circle centered at the origin having radius 10. Find a point Q on the circle whose tangent line at Q passes through the point P = (6, 17).

Thanks for helping!

 

[Gene]

The equation of the circle is
x²+y²=10²
The tangent at P(x,y) has a slope of -x/y
You now have a point-slope equation for the tangent
(y-17)=(-x/y)(x-6)
y²-17y=-x²+6x
y²+x²=6x+17y = 100
y = (100-6x)/17
You can now plug that into the original equation and find x.
There are two answers, one "nice" and one "ugly"

 

[klreed2]

I think I understand where you are getting everything but I have a couple of questions.

What is this equation y = (100-6x)/17 ? I know where you got it, but what is it the equation of?

Also, so there are going to ALWAYS be two tangent lines of a circle that pass through the same point? I'm sure this is a property of a circle, that I'm forgetting so just making sure. Thanks.

 

[Guest]

Also, so there are going to ALWAYS be two tangent lines of a circle that pass through the same point? I'm sure this is a property of a circle, that I'm forgetting so just making sure.

Just try to picture it. Picture a circle and some point outside of the circle. There are two lines you can draw that go through the point and just touch the circle: one on each side of the circle, right?

If the point is outside of the circle, then there are exactly two lines. If the point is on the circle, then there is exactly one line. If the point is inside the circle, then there are exactly zero lines.

 

[klreed2]

ok thanks... but still, what is the equation y = (100-6x)/17 of?

 

[tkhunny]

It is the Tangent Line at Q.

 

[klreed2]

That's what I thought, but I thought there was more than just one tangent line. Also, if this was the tangent line at Q, it must pass through P as well, but when putting coordinates of P (6, 17) into equation - it doesn't work! Am I missing something? Thanks!

 

[Gene]

Sorry, I skipped some things.
y = (100-6x)/17
is the equation of a line thru the two points Q where the tangent lines touch the circle. When you plug that into
x²+y²=10²
you get two points Q = P(x,y) which you can use to get the equations of the two tangent lines. Use the two points form.

 

[Gene]

I'm typing faster than I am thinking. Okay, you have
two Q = P(x,y)s and
P(6,17) and
y-17 =m(x-6)
The two ms for the two tangent equations are (17-y)/(6-x) with x & y from each of the two Q(x,y)s.
It seemed so clear when I wrote it