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Question

Answers

R: Radical center of three circles is the point of concurrence of the radical axes of the circles taken in pairs.

A) Both A and R are true and R is the correct explanation of A

B) Both A and R are true and R is not the correct explanation of A

C) A is true but R is false

D) A is false but R is true

Answer
Verified

Let two circles be

\[S:{x^2} + {y^2} + 2gx + 2fy + c = 0\;\] and

\[S\prime :{x_2} + {y_2} + 2g\prime x + 2f\prime y + c\prime = 0\]

then equation of radical axis is

\[S - S\prime = 0\]

i.e. \[\;2(g - g\prime )x + 2(f - f\prime ) + (c - c\prime ) = 0\]

Radical Centre definition: -

The radical axis of three circles, taken in pairs, meet at a point, which is called their radical centre. The coordinate of radical centre can be found by solving the equation

\[{S_1} = {S_2} = {S_3} = 0\]

Let:

Radical axis of first circle\[{S_1} = {x^2} + {y^2} = 4............................(1)\]

Radical axis of second circle \[{S_2} = {x^2} + {y^2} - 3x = 4....................(2)\]

Radical axis of third circle \[{S_3} = {x^2} + {y^2} - 4y = 4....................(3)\]

Radical axis of two circles is the locus of a point, which moves such that length of tangent drawn from it to the two circles are equal.

Equation of radical axis is${S_1} - {S_2} = 0$. Hence,

$

{S_1} - {S_2} = 0;\; \\

{S_2} - {S_3} = 0;\; \\

{S_3} - {S_1} = 0 \\

$

We will get the equation of radical axis and by solving them we can obtain the required radical center. Radical axis is perpendicular to the straight line joining the centre of circle. So, two circles are non-concentric.

Radical axis of first and second circle is given by i.e. Subtracting equation (1) and (2)

\[

\Rightarrow \left( {{x^2} + {y^2}} \right) - \left( {{x^2} + {y^2} - 3x} \right) = 0 \\

\Rightarrow 3x = 0 \\

\Rightarrow x = 0..............................(4) \\

\]

Radical axis of second and third circle is given by i.e. Subtracting equation (2) and (3)

$

\Rightarrow \left( {{x^2} + {y^2} - 3x} \right) - \left( {{x^2} + {y^2} - 4y} \right) = 0 \\

\Rightarrow 3x + 4y = 0..............................(5) \\

$

Radical axis of third and second circle is given by i.e. Subtracting equation (3) and (1)

$

\Rightarrow \left( {{x^2} + {y^2} - 4y} \right) - \left( {{x^2} + {y^2}} \right) = 0 \\

\Rightarrow - 4y = 0 \\

\Rightarrow y = 0..............................(6) \\

$

Thus the radical centre is (0, 0). Hence A is correct.

Substituting x in equation (5) by equation (4) gives the value of y similar to given by equation (6). These three straight lines meet at a point; since it is obvious that the values of x and y which simultaneously satisfy two of the equations, is also satisfying the third equation.

Hence R is also correct.

So Option (A) is the correct answer.

Similarly, we can use radical center to prove the concurrency of lines. The strategy is to show that each of these lines is the radical axis of a pair of circles and then conclude that they meet at the radical center.