Equation of Circle with Origin as Center

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

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

Q. Let AB be a chord of the circle $x^2+y^2=r^2$ subtending a right angle at the center. Then, the locus of the centroid of triangle PAB as P moves on the circle is

1. a pair of straight lines
2. a parabola
3. a circle
4. an ellipse

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 circle $x^2+y^2=a^2$ intersects the hyperbola $xy=c^2$ at four points
$P(x_1,y_1),Q(x_2,y_2),R(x_3,y_3),andS(x_4,y_4)$, then

1. $x_1+x_2+x_3+x_4=0$
2. $y_1+y_2+y_3+y_4=0$
3. $x_1{x}_2{x}_3{x}_4=c^4$
4. $y_1{y}_2{y}_3{y}_4=c^4$

Q. The number of points with integral coordinates that are interior to the circle $x^2+y^2=16$ is

1. 49
2. 45
3. 43
4. 51

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