Coordinate Geometry - Length of a Line Segment

Coordinate Geometry - Length of a Line Segment

A circle \(L\) is given by the equation \(4{{x}^{2}}+4{{y}^{2}}+8x-16y-29=0\).

  1. Find the coordinates of the centre of the circle and the radius of the circle.
  2. Determine, with working, if the point \(\left( -2,5 \right)\) lies inside or outside the circle.

Another circle \(M\), with centre \(\left( 4,14 \right)\), touches the circle \(L\) at one point. Find

  1. the distance between the two centres.
  2. all the possible equations of circle \(M\) in the form of \({{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\).

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Answer:(a) Centre \((-1,2)\), radius \(=\frac72\) (b) Inside circle \(L\) (c) \(13\text{ units}\) (d) \(x^2-8x+y^2-28y+\frac{487}{4}=0\) or \(x^2-8x+y^2-28y-\frac{241}{4}=0\)

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