General Equation of a Circle

Q.

Find the equation of the circle which passes through the origin and cuts off intercepts $a$ and $b$ respectively from $x$ and $y$-axis.

Q.

Find the equation of the circle which passes through the points (2, 3) and (4, 5) and the centre lies on the straight line y - 4x + 3 = 0.

Q.

Find the equation of the circle which passes through (3, - 2), (-2, 0) and has its centre on the line 2x -y = 3

Q.

Show that the pooints (5, 5), (6, 4), (-2, 4) and (7, 1 ) all lie on a circle, and find its equation, centre and radius.

Q.

Find the equation of the circle which passes through the points (3, 7), (5, 5) and has its centre on the line x - 4y = 1.

Q.

Find the equation of the circle passing through the points :
(i) (5, 7), (8, 1) and (1, 3)
(ii) (1, 2), (3, -4), and (5, -6)
(iii) (5, -8), (-2, 9) and (2, 1)
(iv) (0, 0), (-2, 1) and (-3, 2)

Q.

Find the equation of a circle.
(i) which touches both the axes at a distance of 6 units from the origin.
(ii) which touches x-axis at a distance 5 from the origin and radius 6 units.
(iii) which touches both the axes and passes through the point (2, 1).

(iv) passing through the origin, radius 17 and ordinate of the centre is -15.

Q.

Show that the points (3, -2), (1, 0), (-1, -2) and (1, -4) are concylic.

Q.

If the point (2, k) lies outside the circles
$x^2+y^2+x-2y-14=0$ and $x^2+y^2=13$
then k lies in the interval

1. $(-3,-2)\cup (3,4)$

2. $-3,4$

3. $(-\infty ,-3)\cup (4,\infty )$

4. $(-\infty ,-2)\cup (3,\infty )$

Q. Write the vector equation of the straight line passing through the point $(a,b,c)$ and parallel to y-axis.

Q.

Find the equation of the circle which circumscribes the triangle formed by the lines:
(i) $x+y+3=0,x-y+1=0$ and $x=3$
(ii) $2x+y-3=0,x+y-1=0$ and $3x+2y-5=0$
(iii) $x+y=2,3x-4y=6$ and $x-y=0.$
(iv) $y=x+2,3y=4x$ and $2y=3x$.

Q. Let $C_1:x^2+y^2=1;C_2:(x-10{)}^2+y^2=1$ and $C_3:x^2+y^2-10x–42y+457=0$ be three circles. A circle C has been drawn to touch circles $C_1$ and $C_2$ externally and $C_3$ internally. Now circles $C_1,C_2$ and $C_3$ start rolling on the circumference of circle C in anticlockwise direction with constant speed. The centroid of the triangle formed by joining the centres of rolling circles $C_1,C_2$ and $C_3$ lies on

1. $x^2+y^2-8x-20y+64=0$
2. $x^2+y^2-10x-24y+144=0$
3. $x^2+y^2-4x-2y-40=0$
4. $x^2+y^2-12x-2y=0$

Q. The locus of the centre of a circle, which touches externally the circle $x^2+y^2-6x-6y+14=0$ and also touches the y-axis, is given by the equation

1. $x^2-10x-6y+14=0$
2. $x^2-6x-10y+14=0$
3. $y^2-6x-10y+14=0$
4. $y^2-10x-6y+14=0$

Q. The radius of the circle of least size that passes through $(-2,1)$ and touches both axes is?

1. $3$
2. $1$
3. $5$
4. $2$

Q. The locus of the centre of a circle, which touches externally the circle $x^2+y^2-6x-6y+14=0$ and also touches the y-axis, is given by the equation

1. $x^2-6x-10y+14=0$
2. $x^2-10x-6y+14=0$
3. $y^2-6x-10y+14=0$
4. $y^2-10x-6y+14=0$

Q. The points in the $set\{zϵC:arg\left(\frac{z-2}{z-6i}\right)=\frac{\pi }{2}\}$ lie on the curve which is a (where C denotes the set of all complex numbers) :

1. Parabola
2. Circle
3. Hyperbola
4. Pair of Straight lines