Equation of Family of Circles Passing through Point of Intersection of Two Circles

Q. The equation of the circle passing through the point (–2, 4) and through the points of intersection of the circle $x^2+y^2-2x-6y+6=0$ and the line 3x + 2y - 5 = 0, is

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

Q.

Find the equation of the circle circumscribing the triangle formed by the lines $x+y=0,2x+y=4$ and $x+2y=5$

1. $x^2+y^2+17x+19y+50=0$

2. $x^2+y^2+17x-11y+30=0$

3. $x^2+y^2-17x-11y+30=0$

4. $x^2+y^2-17x-19y+50=0$

Q. Consider a family of circles which are passing through the point (-1, 1) and are tangent to the x-axis. If (h, k) are the coordinates of the centre of the circles, then the set of values of k is given by the interval

1. $0<k<\frac{1}{2}$
2. $k\ge \frac{1}{2}$
3. $-\frac{1}{2}\le k\le \frac{1}{2}$
4. $k\le \frac{1}{2}$

Q.

Find the equation of the circle circumscribing the triangle formed by the lines $L_1=0,L_2=0$and $L_3=0$

1. $L_1+\lambda L_2+\mu L_3=0$

2. $L_1{L}_2+\lambda L_2{L}_3+\mu L_3{L}_1=0$

3. $L_1{L}_2{L}_3=0$

4. $L_1^2+\lambda L_2^2+\mu L_3^2=0$

Q.

Find the equation of family of circles through the intersection of $x^2+y^2-6x+2y+4=0$ and $x^2+y^2+2x-4y-6=0$ whose center lies on $y=x.$

1. $x^2+y^2-\frac{10}{7}x-\frac{10}{7}y-\frac{12}{7}=0$

2. $x^2+y^2+\frac{10}{7}x+\frac{10}{7}y-\frac{12}{7}=0$

3. $x^2+y^2-\frac{10}{7}x+\frac{10}{7}y+\frac{12}{7}=0$

4. $x^2+y^2+\frac{10}{7}x+\frac{10}{7}y+\frac{12}{7}=0$

Q.

The equation of the circle through the points of intersection of $x^2+y^2-1=0,x^2+y^2-2x-4y+1=0$ and touching the line x + 2y = 0, is

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

2. $3(x^2+y^2)+4x-4y-3=0$

3. $x^2+y^2-x-2y=0$

4. $3(x^2+y^2)+4(x+y)-3=0$