Equation of Circle with Origin as Center

Q. Let C be the circle with centre (0, 0) and radius 3 units. The equation of the locus of the midpoints of the chords of the circle C that subtend an angle of $\frac{2\pi }{3}$ at its centre is -

1. 
2. 
3. 
4. 

Q. If the chord y = mx + 1 of the circle $x^2+y^2=1$ subtends an angle of measure ${45}^{\circ }$ at the major segment of the circle then value of m is

1. 2
2. -2
3. -1
4. none of these

Q. The centre of the circle passing through (0, 0) and (1, 0) and touching the circle $x^2+y^2=9$ is

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

Q. For all values of $\theta$, the locus of the point of intersection of the lines $xcos\theta +ysin\theta =a$ and $xsin\theta -ycos\theta =b$ is

1. An ellipse
2. A circle
3. A parabola
4. A hyperbola