Angle between Pair of Tangents

Q. Angle between tangents drawn to $x^2+y^2-2x-4y+1=0$ at the points where it is cut by the line $y=2x+c,is\frac{\pi }{2}$ then

1. $|c|=\sqrt{5}$
2. $|c|=2\sqrt{5}$
3. $|c|=\sqrt{10}$
4. $|c|=2\sqrt{10}$

Q. The angle between the pair of tangents drawn from a point P to the circle $x^2+y^2+4x-6y+9$$si{n}^2\alpha +13co{s}^2\alpha =0$ is $2\alpha$. Then the equation of the locus of the point P is

1. $x^2+y^2+4x-6y+4=0$
2. $x^2+y^2+4x-6y-9=0$
3. $x^2+y^2+4x-6y-4=0$
4. $x^2+y^2+4x-6y+9=0$

Q. Let $2x^2+y^2-3xy=0$ be the equation of a pair of tangents drawn from the origin $O$ to a circle of radius $3$ units with centre in the first quadrant. If $A$ is one of the points of contact, then the length of $OA$ is

1. $3+\sqrt{10}$ units
2. $9(1+\sqrt{10})$ units
3. $3(3+\sqrt{10})$ units
4. $9+\sqrt{10}$ units

Q. Let $2x^2+y^2-3xy=0$ be the equation of a pair of tangents drawn from the origin $O$ to a circle of radius $3$ units with centre in the first quadrant. If $A$ is one of the points of contact, then the length of $OA$ is

Q. The range of values of $a$ such that the angle $\theta$ between the pair of tangents drawn from $(a,0)$ to the circle $x^2+y^2=1$ satisfies $\frac{\pi }{2}<\theta <\pi$, lies in

1. $(-1,1)$
2. $(1,\sqrt{2})$
3. $(-\sqrt{2},-1)$
4. $(-\sqrt{2},\sqrt{2})$