Consider the circles C1 ≡x^2+y^2-4 x+6 y-2=0 C2 ≡x^2+y^2+9 x+6 y-2=0 Statement 1 The centre of t

step 1 So in this question, two circles C1 and C2 are given. In this question we will use the concept o

Consider the circles C1 ≡x^2+y^2-4 x+6 y-2=0 C2 ≡x^2+y^2+9...

Consider the circles $C_{1} \equiv x^{2}+y^{2}-4 x+6 y-2=0$ $C_{2} \equiv x^{2}+y^{2}+9 x+6 y-2=0$ Statement 1 The centre of the circle passing through the intersection of the circles $\mathrm{C}_{1}=0$ and $\mathrm{C}_{2}=0$ and through the point $(1,1)$ lies on the line $3 x+y=0$ and Statement 2 Any circle through the intersection of $\mathrm{C}_{1}=0$ and $\mathrm{C}_{2}=0$ can be written as $\mathrm{C}_{1}+\lambda \mathrm{C}_{2}=0$ where, $\lambda$ is a constant.

Consider the circles
$C_{1} \equiv x^{2}+y^{2}-4 x+6 y-2=0$
$C_{2} \equiv x^{2}+y^{2}+9 x+6 y-2=0$
Statement 1
The centre of the circle passing through the intersection of the circles $\mathrm{C}_{1}=0$ and $\mathrm{C}_{2}=0$ and through the point $(1,1)$
lies on the line $3 x+y=0$
and
Statement 2 Any circle through the intersection of $\mathrm{C}_{1}=0$ and $\mathrm{C}_{2}=0$ can be written as $\mathrm{C}_{1}+\lambda \mathrm{C}_{2}=0$ where, $\lambda$ is a constant.

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Key Concepts

Pencil of Circles

A pencil of circles is a family of circles that all pass through the same two fixed points, which are the intersections of two given circles. Any circle in this family can be represented as a linear combination of the equations of the two fixed circles (usually written in the form C? + ?C? = 0), which characterizes all circles that share these common intersection points.

Standard Equation of a Circle

The standard equation of a circle in the plane is written as x² + y² + 2gx + 2fy + c = 0, where the center of the circle is located at (–g, –f) and the radius is given by ?(g² + f² – c). Converting a circle's equation to this standard form is essential for extracting its geometric properties, such as the center, radius, and for analyzing its position relative to other circles or lines.

Radical Axis

The radical axis of two circles is defined as the set of points that have equal power with respect to both circles. It is typically obtained by subtracting the equations of the circles to form a linear equation. This line plays a significant role in circle geometry, especially in problems involving the intersections of circles and the properties of the centers of circles drawn through those intersection points.

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