Deduce equations of tangents

Jul 2009
338
14
Singapore
Deduce the equations of the tangents to the circle from the point (1,0).
The equation of the circle \(\displaystyle (x-2)^2+(y-2)^2=9\)
my problem is that the are no tangents to the circle through the point. I got the polar of the point, \(\displaystyle 2y+x+3=0\) but i don't know how to continue.
Thanks
 
Jun 2009
806
275
Let y = mx + c is the equation of the tangents.

It passes from (1,0). So m = -c.

Condition for a straight line to be tangent to a circle of radius a is

\(\displaystyle c^2 = (1+m^2)a^2 \)

So \(\displaystyle m^2 = (1+m^2)a^2\)

Here the value of m is imaginary. So there is no tangent from the given point.
 
Jul 2009
338
14
Singapore
This is the full question:
Find the equation of the pair of lies that pass through the origin and through the points of intersection of the line \(\displaystyle y=\lambda(x+1)\) and the circle \(\displaystyle (x-2)^2+(y-2)^2=9\). If these lines are perpendicular find \(\displaystyle \lambda\). For what value of [/tex]\lambda[/tex] do these lines coincide? Deduce the equations of the tangents to the circle from the point (1,0).
This is the last part. I've done all the rest. For when the lines coincide i do have a small problem.
The equation of the line pair is:
\(\displaystyle y^2(\lambda^2-4\lambda-1)+2\lambda xy(2\lambda -1)+4\lambda^2x^2=0\)
If the lines coincide then \(\displaystyle h^2-ab=0\)
\(\displaystyle \lambda^2(2\lambda -1)^2-(\lambda^2-4\lambda-1)(4\lambda^2)=0\)
I worked it out and found \(\displaystyle \lambda=0, -\frac{5}{12}\)
Answer says \(\displaystyle \lambda=\infty, -\frac{5}{12}\)
These two are the problems I have
 
Jun 2009
806
275
The given line passes through a fixed point ( -1, 0).

Two lines through the origin will co-inside when the given line is either along the x-axis or tangent at (-1, 0).
 
Jul 2009
338
14
Singapore
umm i don't exactly understand what you mean
 
Jun 2009
806
275
umm i don't exactly understand what you mean
What I mean is that, the given straight line passes through a fixed point (-1, 0). Two lines from the origin will co-inside only when the given line passes through the origin.

The point (1, 0) lies inside the circle. So you cannot draw a tangent from that point to the circle. You have to check the problem.
 
Jul 2009
338
14
Singapore
I copied it exactly from the book. verbatim