`(2, 1)` is the points of intersection of two lines

A circle (with centre at O) is touching two intersecting lines AX and BY. The two points of contact A and B subtend an angle of 65^(@) at any point C on the circumference of the circle. If P is the point of intersection for the two lines, then the measure of angle APO is :

The lines y = 2x + sqrt(76) and 2y + x = 8 touch the ellips (x^(2))/(16) +(y^(2))/(12) = 1 . If the point of intersection of these two lines lie on a circle. Whose centre coincides with the centre of that ellipse, then the equation of that circle is

The lines y = 2x + sqrt 76 and 2y + x=8 touch the ellipse (x ^(2))/(16 )+(y ^(2))/(12)=1. If the point of intersection of these two lines lie on a circle, whose centre coincides with the centre of that ellipse, then the equation of that circle is

The lines y = 2x + sqrt 76 and 2y + x=8 touch the ellipse (x ^(2))/(16 )+(y ^(2))/(12)=1. If the point of intersection of these two lines lie on a circle, whose centre coincides with the centre of that ellipse, then the equation of that circle is

The distance between the point of intersection of the two lines 2009x^(2)+2010xy+2011y^(2)=0 and the point (1,1) is

A variable straight line through the point of intersection of the two lines (x)/(3)+(y)/(2)=1and(x)/(2)+(y)/(3)=1 meets the coordinate axes at A and B . Find the locus of the middle point of AB.