NB: Everything I know about a pair of straight lines is from this PDF
. I have made use of quite a few formulas given in the PDF in my answer.
Q: Find the equations of the tangents from the point $A(3,2)$ to the circle $x^2+y^2=4$ and hence find the angle between the pair of tangents.
A: I started off by applying the rule T^2=SS1 (i.e the joint equation of a pair of tangents drawn from a point $A(x1,y1)$ to the circle).
After simplifying the expression, I got
$$5y^2+24x+16y-12xy-52=0$$
Approach 1: By using the formula for the angle between a pair of straight lines(page 40 of the given PDF), I got the value of tan $B =\frac{24}{5} \space radians $ which is the required angle.
Approach 2: Assuming that the tangent touches the circle at P, then OP=radius=2 and AP= length of tangent =square root of S1= 3.
Then further solving using trigonometry, I get tan $B=\frac{12}{5} \space radians$.
Why am I getting different answers? The actual answer is the one from my 2nd approach.But how is my second approach wrong?
Also,someone might also want to look at the following PDF for information on theory of tangents of circles
Best Answer
one of the tangents from $(3,2)$ touch the circle at $(0,2).$ if you look at the right angle triangle formed by the points $(0,0), (3,2)$ and $(0,2),$ then half the angle between the tangents is $\arctan(2/3).$ now use the formula $\tan(2t) = \frac{2\tan t}{1- \tan^2 t} = \frac{4/3}{1-4/9} = \frac{12}5.$