Locus of the foot of the perpendiculars drawn from the...

Locus of the foot of the perpendiculars drawn from the centre of the circle $x^{2}+y^{2}+2 g x+2 f y+2 c=0$ to its chords which subtend a right angle at the centre is (a) $(x+g)^{2}+(y+f)^{2}=f^{2}+g^{2}-c$ (b) $(x+g)^{2}+(y+f)^{2}=\frac{c}{2}$ (c) $(x+g)^{2}+(y+f)^{2}=2\left(f^{2}+g^{2}-c\right)$ (d) $(x+g)^{2}+(y+f)^{2}=\frac{f^{2}+g^{2}-c}{2}$

Locus of the foot of the perpendiculars drawn from the centre of the circle $x^{2}+y^{2}+2 g x+2 f y+2 c=0$ to its chords which subtend a right angle at the centre is
(a) $(x+g)^{2}+(y+f)^{2}=f^{2}+g^{2}-c$
(b) $(x+g)^{2}+(y+f)^{2}=\frac{c}{2}$
(c) $(x+g)^{2}+(y+f)^{2}=2\left(f^{2}+g^{2}-c\right)$
(d) $(x+g)^{2}+(y+f)^{2}=\frac{f^{2}+g^{2}-c}{2}$

Key Concepts

Chord Subtending a Fixed Angle at the Center

In circle geometry, any chord that subtends a fixed angle at the center has a fixed perpendicular distance from the center. This distance can be calculated using the relation d = r cos(?/2), where r is the radius of the circle and ? is the central angle. In the case of a 90° angle, this results in a constant distance of r/?2 from the center to the chord.

Locus Defined by a Constant Distance from a Point

A locus is the set of all points satisfying a given condition. When the condition is that the points lie at a fixed distance from a specific point, the locus is a circle. In this problem, since the foot of the perpendicular from the center to any chord subtending a right angle must be at a constant distance from the center, its locus is itself a circle.

Conversion to Center-Radius Form of a Circle

The general equation of a circle can be rewritten in the center-radius form by completing the square in both x and y. This process reveals the coordinates of the centre and the length of the radius explicitly, which is critical when deducing geometrical relationships such as distances between points or loci derived from the circle.

Perpendicular from a Point to a Line

The concept of the foot of the perpendicular from a point to a line involves finding the point on the line that is closest to the given point. In this context, drawing the perpendicular from the centre to a chord identifies a unique point whose distance from the centre (the perpendicular distance) is used to derive the locus of such points when the chord subtends a fixed angle at the centre.

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