Tangents on a circle

We can draw the following diagram as per the given question :

Now, taking $$\triangle AOT$$ and $$\triangle AOQ$$

AO is the common side

AT=AQ (Tangents drawn from an external point are equal in length)

OT=OQ (radius of circle)

So, by S.S.S., $$\triangle\ AOT$$ is congruent to $$\triangle\ AOQ$$

So, by C.P.C.T., $$\angle\ AOT=\angle\ AOQ$$

Similarly, for $$\triangle\ BOT\ $$ and $$\triangle\ BOR\ $$, we can do the same thing,

So, we can say $$\triangle\ BOT$$ and $$\triangle\ BOR$$ will also be congruent

Now, $$\angle\ QOR=\angle\ AOQ+\angle\ AOB+\angle\ ROB=\angle\ AOT+\angle\ AOB+\angle\ BOT=2\angle\ AOB=2\times\ 50^{\circ\ }=100^{\circ\ }$$

Now, in quadrilateral OQPR,

$$\angle\ OQP=\angle\ ORP=90^{\circ\ }$$ (Tangents drawn to a point is perpendicular to the radius drawn at the same point)

Now, sum of angles of a quadrilateral is $$360^{\circ\ }$$

So, $$\angle\ APB+\angle\ QOR=360^{\circ\ }-\left(90^{\circ\ }+90^{\circ\ }\right)=180^{\circ\ }$$

So, $$\angle\ APB=180^{\circ\ }-100^{\circ\ }=80^{\circ\ }$$

$$x^{2}$$ + $$y^{2}$$ = 233 and x - y = 5. So, x = 13 cm and y = 8 cm

From a point outside of the circle, minimum 0 tangents can be drawn and a maximum of 2 tangents can be drawn. Therefore, there are three cases possible. 

AB is tangent - Only one intersection point, i.e. B

AD is secant - Two intersection points C and D

Case 1: Number of tangents is zero.

This implies 1001 lines drawn from a point are secants. There are two points of intersection on a circle if a secant passes through it. Therefore, total number of points of intersection in this case is 2*1001 = 2002

Case 2: Number of tangents is one and remaining 1000 lines are secants.

A tangent will have only one point of intersection on circle whereas a secant has two points of intersection. Therefore, total number of points of intersection in this case is 1 + 2*1000 = 2001

Case 3: Number of tangents is two and remaining 999 lines are secants.

A tangent will have only one point of intersection on circle whereas a secant has two points of intersection. Therefore, total number of points of intersection in this case is 12 + 2*999 = 2000

Therefore, the possible remainders are 2002%6 = 4, 2001%6 = 3, 2000%6 = 2

The answer is option C.