Draw the unit circle with center (0,0)and plot the point P=(6,2). Observe there are TWO lines tangent to the circle passing through the point P.

Question

1)the tangent line on top of the circle(L1) has coordinates?

a) what is the equation for L1?

2) the tangent line below the circle(L2) has coordinates?

b)what is the equation for L2?

I have the equation for the circle but after that I get confused because I cannot find the slope of the tangent lines and i know the line from the center of the circle to the point of tangent touching the circle is perpendicular.. For L1 i get to the point of y-2=m(x-6). once I get to that point I am completely lost.

Answer 1

Hi Jon,

        I'll tell you the approach I took:

First I made a rough to scale sketch of the circle and the two lines.

Then write the equation for the line passing through P(6,2) y-2=m(x-6) m=slope

write the equation for the line passing through (0,0) and perpendicular to the first line y=-(1/m)x

If you solve for m we have m=-y/x and sub in the first equation we have y-2=-(y/x)(x-6) now y and x have to be on the unit circle so x²+y²=1 if you solve these two equations for x and y  (messy) I got x=-.16225, y=.98675 and x=.46225, y=-.88675.

 

The first set are the coordinates of the point of tangency on the top and the second set are the corresponding lower point.

Now since you have two point on each line it's easy to write the equations for the lines or you can calculate the slope m directly. 

 

This was a nice problem a little out of the ordinary

 

Hope this helps

Jim