Equation of Circle with (h,k) as Center

Q.

How do you find the centre of a circle given two points $?$

Q. The lines 2x - 3y = 5 and 3x - 4y = 7 are the diameters of a circle of area 154 square units. The equation of the circle is

1. $x^2+y^2+2x-2y=62$
2. $x^2+y^2-2x+2y=47$
3. $x^2+y^2+2x-2y=47$
4. $x^2+y^2-2x+2y=62$

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. Equation of a circle whose centre is origin and radius is equal to the distance between the lines x = 1 and x = -1 is

1. $x^2+y^2=1$
2. $x^2+y^2=\sqrt{2}$
3. $x^2+y^2=4$
4. $x^2+y^2=-4$

Q. The equation of the circle whose radius is 5 and which touches the circle $x^2+y^2-2x-4y-20=0$ externally at the point (5, 5), is

1. $x^2+y^2-18x-16y-120=0$
2. $x^2+y^2-18x-16y+120=0$
3. $x^2+y^2+18x+16y-120=0$
4. $x^2+y^2+18x-16y+120=0$

Q. The circle passing through $(1,-2)$ and touching the axis of $x$ at $(3,0)$ also passes through the point

1. $(-5,2)$
2. $(2,-5)$
3. $(5,-2)$
4. $(-2,5)$

Q. A circle passes through $(-2,4)$ and touches the $y-$axis at $(0,2).$ Then which of the following equations can represent a diameter of the circle?

1. $4x+5y-6=0$
2. $2x-3y+10=0$
3. $3x+4y-3=0$
4. $5x+2y+4=0$

Q. The line $4x+3y-4=0$ divides the circumference of the circle centred at $(5,3),$ in the ratio $1:2.$ Then the equation of the circle is

1. $x^2+y^2-10x-6y-66=0$
2. $x^2+y^2-10x-6y+100=0$
3. $x^2+y^2-10x-6y+66=0$
4. $x^2+y^2-10x-6y-100=0$

Q. A circle which made an intercept of $2a$ on $y-$axis and $x-$axis is the tangent to it. The locus of the center of the circle is

1. Hyperbola with eccentricity $\sqrt{2}$
2. Hyperbola with eccentricity $\sqrt{3}$
3. Ellipse with eccentricity $\frac{1}{\sqrt{2}}$
4. Parabola with focus $(0,2a)$

Q. The equation of the circle whose centre is $(-5,4)$ and passes through the origin, is

1. $x^2+y^2-8x+10y=0$
2. $x^2+y^2+8x-10y+12=0$
3. $x^2+y^2+10x-8y=0$
4. $x^2+y^2+10x-8y+25=0$

Q. Let $x,y$ be real variables satisfying the $x^2+y^2+8x-10y-40=0$. Let $a=max\left\{\sqrt{(x+2{)}^2+(y-3{)}^2}\right\}$ and $b=min\left\{\sqrt{(x+2{)}^2+(y-3{)}^2}\right\}$, then

1. $a+b=18$
2. $a+b=\sqrt{2}$
3. $a-b=4\sqrt{2}$
4. $a.b=73$

Q. Equation of the circle inscribed in $|x+1|+|y-6|=2$ is

1. $(x-1{)}^2+(y-6{)}^2=2$
2. $(x-1{)}^2+(y+6{)}^2=2$
3. $(x+1{)}^2+(y-6{)}^2=2$
4. $(x+1{)}^2+(y+6{)}^2=2$

Q. The locus of the centres of the circles, which touch the circle, $x^2+y^2=1$ externally, also touch the $y-$axis and lie in the first quadrant, is:

1. $y=\sqrt{1+2x},x\ge 0$
2. $y=\sqrt{1+4x},x\ge 0$
3. $x=\sqrt{1+4y},y\ge 0$
4. $x=\sqrt{1+2y},y\ge 0$

Q. 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, is

1. $x^2+y^2+4x-10y+25=0$
2. $x^2+y^2-4x-10y+25=0$
3. $x^2+y^2-4x-10y+16=0$
4. $x^2+y^2-14y+8=0$

Q.

The circle passing through (1, -2) and touching the axis of x at (3, 0) also passes through the point

1. (-5, 2)

2. (2, -5)

3. (5, -2)

4. (-2, 5)

Q.

Let $c_1:x^2+y^2=1;C_2:(x-10{)}^2+y^2=1and{C}_3;x^2+y^2-10x-42y+457=0$ be three circle.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. $x^2+y^2-12x-22y+144=0$

2. $x^2+y^2-10x-24y+144=0$

3. $x^2+y^2-8x-20y+64=0$

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

Q.

Let $a,b,c,d$ be real numbers such that $\{\begin{array}{c}{a}^2+b^2+2a-4b+4=0\\ c^2+d^2-4c+4d+4=0\end{array}.$ Let $m$ and $M$ be the minimum and the maximum values of $(a-c{)}^2+(b-d{)}^2,$ respectively. The value of $m\times M$ is
(correct answer + 3, wrong answer 0)