Properties of Radical Axis

Q. The radical centre of three circles described on the three sides $4x-7y+10=0,x+y-5=0$ and $7x+4y-15=0$ of a triangle as diameters.

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

Q. Centre of circle $(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0$ is

1. 
2. 
3. 
4. 

Q. Circles are drawn through the points $(a,b)$ and $(b,-a)$ such that common chord substend an angle of $45°$ on the circumference on any of the circles. If distance between the centres is $\sqrt{k}$ times the radius of the smaller circle , then $k=$

Q.

Which of the following statements is/are correct?

1. The radical axis of two circles is the locus of points whose power with respect to the two circles is equal.

2. The common point of intersection of the radical axes of three circles taken two at a time called the radical center of three circles.

1. Only 2

2. Only 1

3. Both 1 and 2

4. None of these

Q.

If (h, k) is a point from which the tangents to the three circles $x^2+y^2-4x+7=0$, $2x^2+2y^2-3x+5y+9=0$ and $x^2+y^2+y=0$ are equal in length. Find the value of h + k ___ .

Q. Consider three circles whose equations are $x^2+y^2+3x+2y+1=0$, $x^2+y^2-x+6y+5=0$ and $x^2+y^2+5x-8y+15=0$, then

1. equation of the circle which is orthogonal to given circles is$x^2+y^2-6x-4y-14=0$
2. equation of the circle which is orthogonal to given circles is$x^2+y^2-6x+4y-13=0$
3. radical centre of the circles is $(3,2)$
4. radical centre of the circles is$(3,-2)$

Q. The radical centre of three circles described on the three sides $4x-7y+10=0,x+y-5=0$ and $7x+4y-15=0$ of a triangle as diameters.

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

Q. If a circle C, whose radius is 3, touches externally the circle, $x^2+y^2+2x–4y–4=0$ at the point $(2,2)$, then the length of the intercept cut by this circle C, on the $X$-axis is equal to :

1. 
   $2\sqrt{3}$
2. $\sqrt{5}$
3. $3\sqrt{2}$
4. $2\sqrt{5}$

Q. Let $S_1$ and $S_2$ be two circles touching externally and having radius as $2$ and $3$ respectively. $S_1$ and $S_2$ touch a variable circle $S_3$ internally at $A$ and $B$ respectively. If the tangents to $S_3$ at $A$ and $B$ meet at $T$ and $TA=4\text{units}$, then which of the following is/are correct?
(Here, $C_i$ represents the centre of circle $S_i$)

1. The radius of circle $S_3$ is $8$ units.
2. The area of circle circumscribing $△TAB$ is $20\pi$ sq. units.
3. $C_3{C}_1+C_3{C}_2=5$
4. $C_3{C}_1–C_3{C}_2=1$

Q.

If (h, k) is a point from which the tangents to the three circles $x^2+y^2-4x+7=0$, $2x^2+2y^2-3x+5y+9=0$ and $x^2+y^2+y=0$ are equal in length. Find the value of h + k ___ .

Q. Given a semicircle of radius $1$, let $a$ be the side of an equilateral triangle which is inscribed in the semicircle with its vertices on the boundary of the semicircle (boundary includes the bounding diameter also). Then the set of possible values of $a$ is

1. $\{1,\frac{2}{\sqrt{3}}\}$
2. $\left\{1\right\}$
3. the set of all positive real numbers not exceeding $\frac{2}{\sqrt{3}}$
4. the set of all real numbers which are greater than or equal to 1, but less than or equal to $\frac{2}{\sqrt{3}}$

Q. If a circle C, whose radius is 3, touches externally the circle, $x^2+y^2+2x–4y–4=0$ at the point $(2,2)$, then the length of the intercept cut by this circle C, on the $X$-axis is equal to :

1. 
   $2\sqrt{3}$
2. $3\sqrt{2}$
3. $2\sqrt{5}$
4. $\sqrt{5}$