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GradeRadical Axis of Two Circles

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If (a, c) and (b, c) are the centres of the two circles whose radial axis is y-axis. If the radius of the first circle is r then the diameter of the other circle is

$

\left( {\text{A}} \right)2\sqrt {{r^2} - {b^2} + {a^2}} \\

\left( {\text{B}} \right)\sqrt {{r^2} - {a^2} + {b^2}} \\

\left( {\text{C}} \right)\left( {{r^2} - {b^2} + {a^2}} \right) \\

\left( {\text{D}} \right)2\sqrt {{r^2} - {a^2} + {b^2}} \;

$

$

\left( {\text{A}} \right)2\sqrt {{r^2} - {b^2} + {a^2}} \\

\left( {\text{B}} \right)\sqrt {{r^2} - {a^2} + {b^2}} \\

\left( {\text{C}} \right)\left( {{r^2} - {b^2} + {a^2}} \right) \\

\left( {\text{D}} \right)2\sqrt {{r^2} - {a^2} + {b^2}} \;

$

Two circles of equal radiuses a cut orthogonally. If their centres are $(2,3)$ and $(5,6)$, then the radical axis of these circles passes through the point.

A. $(3a,5a)$

B. $(2a,a)$

C. $\left( {a,\dfrac{{5a}}{3}} \right)$

D. $(a,a)$

A. $(3a,5a)$

B. $(2a,a)$

C. $\left( {a,\dfrac{{5a}}{3}} \right)$

D. $(a,a)$

If P and Q are the points of intersection of the circles ${x^2} + {y^2} + 3x + 7y + 2p - 5 = 0{\text{ and }}{x^2} + {y^2} + 2x + 2y - {p^2} = 0$, then there is a circle passing through P, Q and (1, 1) for

$

{\text{A}}{\text{. all values of p}}{\text{.}} \\

{\text{B}}{\text{. all except one value of p}}{\text{.}} \\

{\text{C}}{\text{. all except two values of p}}{\text{.}} \\

{\text{D}}{\text{. exactly one value of p}}{\text{.}} \\

$

$

{\text{A}}{\text{. all values of p}}{\text{.}} \\

{\text{B}}{\text{. all except one value of p}}{\text{.}} \\

{\text{C}}{\text{. all except two values of p}}{\text{.}} \\

{\text{D}}{\text{. exactly one value of p}}{\text{.}} \\

$

A: The radical center of the circles ${x^2} + {y^2} = 4,\;{x^2} + {y^2} - 3x = 4,\;{x^2} + {y^2} - 4y = 4$ is (0, 0)

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

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

$LO{{L}^{'}},MO{{M}^{'}}$ are two chords of a parabola passing through a point O on its axis. Prove that the radical axis of the circles described on $L{{L}^{'}},M{{M}^{'}}$ has diameter through the vertex of the parabola.

A circle passes through the point (3, 4) and cuts the circle ${x^2} + {y^2} = {a^2}$ orthogonally, the locus of its centre is a straight line. If the distance of this straight line from the origin is 25, then ${a^2}$ =

A.250

B.225

C.100

D.25

A.250

B.225

C.100

D.25

Find the radical axis of the pairs of circles ${x^2} + {y^2} - xy + 6x - 7y + 8 = 0$ and ${x^2} + {y^2} - xy - 4 = 0$ , the axes being inclined at ${120^ \circ }$

Define coaxial circles and deduce their equation in simplest form.

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