The length of the common chord of the circles x^2+y^2+6 x=0, x^2+y^2+3 y=0 is (a) (6)/(√(5)) (b)

step 1 In this question, we are given with the equation of two circles, that is x square plus y square

The length of the common chord of the circles x^2+y^2+6 x=0,...

Key Concepts

Completing the Square

This is an algebraic technique used to rewrite a quadratic expression in a form that easily reveals properties of conic sections, especially circles. In the context of circle equations, it transforms a general second-degree polynomial into the standard form (x - h)² + (y - k)² = r², allowing identification of the circle's center and its radius.

Center and Radius of a Circle

Once the equation of a circle is rewritten in standard form via completing the square, the coefficients directly give the coordinates of the center (h, k) and the radius r. Understanding these components is crucial when analyzing the relative positions of circles and their intersections.

Distance Formula

The distance formula, derived from the Pythagorean theorem, calculates the distance between two points in the plane. It is used to determine the separation between the centers of two circles, which is a critical factor in analyzing how the circles intersect and in finding properties like the length of the common chord.

Common Chord of Two Circles

When two circles intersect, they share a line segment called the common chord. The length of this chord can be determined using geometric relations that involve the radii of the circles and the distance between their centers. This concept is important for solving problems related to intersections of circles.

Radical Axis

The radical axis is the line along which the power of a point with respect to two circles is the same. In the case of intersecting circles, the radical axis coincides with the line through the endpoints of the common chord. It offers a useful geometric perspective in understanding the properties and measurement of the chord.

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