50 followers question.

If you have solved this question then you can relax now but if you have not then cool down and let me show you that this question was not a very big deal.

We can not visualize the rough sketch of parabola from the given equation. For that we should first rewrite the equation as $13[(x-4)^2 + (y-5)^2] =(2x+3y-5)^2$

or $\sqrt{(x-4)^2 + (y-5)^2} = \dfrac{|2x+3y-5|}{\sqrt{13}}$

This equation should strike your brain that LHS is distance of a point from a fixed point and RHS is distance of a point from a line.

Hold on a second. Is this the very basic definition of parabola? Yes, it is. Then the point $(4,5)$ must be its focus and the line $2x+3y-5 = 0$ must be its directrix.

Now we can visualize the parabola (although very little). We also have the point from which we have to construct the tangents. Then should we write the equation of tangents and find the points then calculate the measure of the angles? But wait here. Are we that much stupid to go on that very very long path? There is still one small way.

If we take a very closer look on the point $T$ , we would find that this point lies on the directrix itself. This gives us all the necessary data to solve this problem without any further calculation. You are wondering how then lets see ahead.

Point $T$ is the point of intersection of tangents at point $P$ & $Q$ and it lies on the directrix. We all know that in parabola tangents which are perpendicular to each other intersects only on directrix. So $\angle PTQ = \dfrac{\pi}{2}$ . Now you would ask what about the other two angles and my answer to you would be to have some patience. It would also come like the above one.

There is a property of parabola which tells that the triangle formed from the focus of a parabola, the point on the parabola at which the tangent is drawn and the point of intersection of the tangent and the directrix of the parabola is a right angled triangle and right angled at the focus. You would think that is it necessary. Yes it is.

So $\angle TSP = \dfrac{\pi}{2}$ . Similarly $\angle TSQ = \dfrac{\pi}{2}$

Now our jigsaw puzzle is solved. And the correct answer to this question is $3\times \dfrac{\pi}{2} \approx 4.712$ .

Thanks for having patience while reading my solution and also sorry for my mistakes in grammar. I hope you like my problem and its solution.