Official solution
Here PQ is the chord of contact.
Equation of PQ is
x.0 - y.3 = 36
\[ \Rightarrow \,\,\,\,y = - 12\]
Putting value of y = - 12 in the equation
4x2 - y2 = 36
we get , 4x2 - 144 = 36
\[ \Rightarrow \,\,\,\,4{x^2}\, = \,180\]
\[ \Rightarrow \,\,\,\,\,{x^2}\,\, = \,\,45\]
\[ \Rightarrow \,\,\,\,x = \pm \,3\sqrt 5 \]
\[\therefore\,\,\,\] Coordinate of \[P = \left( { - 3\sqrt 5 , - 12} \right)\] and
Coordinate of \[Q = \left( {3\sqrt 5 , - 12} \right)\]
\[\therefore\,\,\,\] Length of PQ \[ = 3\sqrt 5 + 3\sqrt 5 \]
\[ = 6\sqrt 5 .\]
TM is the height of the triangle Length of TM = 12 + 3 = 15
\[\therefore\,\,\,\] Area of \[\Delta PQT\] = \[{1 \over 2} \times 6\sqrt 5 \times 15\]
\[ = 45\sqrt 5 \] sq. units