Problem 57 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(S=x^{2}+y^{2}+2 g x+2 f y+c=0\) is a given circle, then the locus of the foot of the perpendicular drawn from origin upon any chord of \(S\) which subtends a right angle at the origin, is (A) \(2\left(x^{2}+y^{2}\right)+2 g x+2 f y+c=0\) (B) \(2\left(x^{2}+y^{2}\right)+2 g x+2 f y-c=0\) (C) \(x^{2}+y^{2}+g x+f y+c=0\) (D) none of these

Step-by-Step Solution

Answer

The answer is (A).

Step 1: Equation of the Circle

The given equation of the circle is \(x^{2} + y^{2} + 2gx + 2fy + c = 0\). This can be rewritten in the standard form \((x + g)^2 + (y + f)^2 = r^2\), where \(-g\), \(-f\) is the center and \(r\) is the radius of the circle.

Step 2: Understand Chord Subtending a Right Angle

A chord that subtends a right angle at the origin means that the angle between the lines joining the origin to the endpoints of the chord is \(90^{\circ}\). This also means that the circle passing through the endpoints of the chord has the origin as its center.

Step 3: Geometric Locus Definition

The locus of all points such that the circle with these points as diameters always passes through the origin is sought after. Since the circle subtending a right angle at the origin implies that it has the origin as the center, we solve this by understanding the properties of perpendicular drawn from origin.

Step 4: Solve for Locus of Perpendicular Foot

Apply the property of perpendicular from origin onto the chord which subtends right angle. The perpendicular from origin to any chord \(xy\) can be represented as \( \text{distance}(x, y) = 0\). Rewriting the given circle equation twice we get: \(2(x^2 + y^2) + 2gx + 2fy + c = 0\).

Step 5: Conclusion

The locus of the foot of the perpendicular drawn from the origin upon any chord which subtends a right angle at the origin is \(2(x^2 + y^2) + 2gx + 2fy + c = 0\), which matches option (A).

Key Concepts

Circle Geometry

Circle geometry is a branch of mathematics that studies the properties and relations of points, lines, and figures in the plane centered around circles. A circle can be defined as the set of all points that are equidistant from a fixed point called the center.
A basic circle equation in the Euclidean plane takes the form \(x^{2} + y^{2} + 2gx + 2fy + c = 0\). By completing the square, this can be rewritten to its standard form \((x + g)^2 + (y + f)^2 = r^2\), where the center is \((-g, -f)\) and \(r\) is the radius.
Understanding the geometry of a circle allows us to explore more complex concepts like loci and tangents.

Perpendicularity

Perpendicularity refers to the concept where two lines intersect at a right angle, or 90 degrees. In mathematical terms, two lines are perpendicular if the product of their slopes is -1.
In the context of circle geometry, perpendicularity is crucial when dealing with the concept of tangents, chords, and diameters. For instance, when you draw a perpendicular from the center of a circle to a chord, you are bisecting that chord.
Perpendicular lines help us solve many geometric problems by creating symmetric, recognizable patterns in figures, which in turn allows for easier calculation and understanding.

Chord Subtending Right Angle

A chord subtending a right angle at a point on the circle, specifically at the origin in this exercise, signifies that the endpoints of the chord, along with the origin, form a right triangle.
This idea is deeply related to the fact that, if a circle passes through these three points, the circle considered has the origin as one of its special points, typically its center or an endpoint of its diameter. This property allows unique derivations in problems concerning loci and the positioning of geometric figures.
Understanding this concept is vital in determining relationships within a circle and applying these to solve exercises involving angles and segment distances.

Locus

A locus is a set of points that satisfy specific conditions or rules. In geometry, the term often refers to paths or areas describing the properties or movement of points, like circles, parabolas, or lines.
For the exercise, the locus describes the path of the foot of the perpendicular dropped from the origin to a circle's chord, which subtends a 90-degree angle at the origin. This forms a particular circle, capturing where all such points reside.
Understanding loci helps in visualizing complex shapes and patterns by simplifying them into recognizable forms, which is useful in both theoretical and practical applications.

Circle Equation

The circle equation expresses all points that form a circle with a specific center and radius. Typically, the equation is \(x^{2} + y^{2} + 2gx + 2fy + c = 0\), where \(-g, -f\) represents the center coordinates, and the rearranged equation provides the radius \((r)\).
In the given exercise, manipulating this circle equation helps derive the locus of the chord subtending a right angle at the origin. By understanding this form of representation, one can predict and calculate properties such as position and distance within the context of the circle.
Mastering the circle equation is fundamental in geometry as it underpins the exploration and solution of a vast array of geometrical problems involving circular shapes.