This problem looks pretty hard but can be simplified using transalation of axes to bring the circle’s center at the origin.
We know that the circle’s center lies at (3, 2) and its radius is 3. Hence, we shift the origin to (3, 2) and the equations transform into the following form:
x
2+y
2=3 –
4x-3y=0 –
Now, visualize a point on the line
extending two tangents on the circle
which makes an angle whose tangent is 24/7. Now (as in figure), angle
P
1PP
2 is equally divided into CPP
2 and CPP
1 which can be equal to
.
(as tan
must be positive, due to acuteness)
Now we have found the angle CPP
2 which can be used to find another equation of
.
(radius=3 and length of tangent=
S
1)
On equating the values of tan
, we get another circle on which our required points lie...
and we already know
On solving, you get
Hence, you can now shift back the origin and find the equations of the pairs of tangents using the standard formula pretty easily.
Thanks,
Shukant Pal
