If P=(1,0), Q=(-1,0) and R=(2,0) are three given points,...

If $P=(1,0), Q=(-1,0)$ and $R=(2,0)$ are three given points, then locus of the point $S$ satisfying the relation $S Q^{2}+S R^{2}=2 S P^{2}$, is [1988-2 Marks] (a) a straight line parallel to $x$-axis (b) a circle passing through the origin (c) a circle with the centre at the origin (d) a straigth line parallel to y-axis.

If $P=(1,0), Q=(-1,0)$ and $R=(2,0)$ are three given points, then locus of the point $S$ satisfying the relation $S Q^{2}+S R^{2}=2 S P^{2}$, is
[1988-2 Marks]
(a) a straight line parallel to $x$-axis
(b) a circle passing through the origin
(c) a circle with the centre at the origin
(d) a straigth line parallel to y-axis.

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Key Concepts

Distance Formula

The distance formula, derived from the Pythagorean theorem, is used to compute the distance between two points in the coordinate plane, typically expressed as d = ?((x2 - x1)² + (y2 - y1)²). Squaring the distances simplifies equations by eliminating square roots, which is common in many geometry problems.

Geometric Locus

The concept of a locus refers to the set of all points that satisfy a specific condition or equation. In coordinate geometry, determining a locus involves translating geometric relationships, such as distance relations, into algebraic equations and then identifying the geometric shape (such as a line or circle) represented by that equation.

Algebraic Simplification in Coordinate Geometry

Once geometric conditions are expressed as algebraic equations using the distance formula, the next step is to simplify these equations. This process involves expanding squared terms, combining like terms, and cancelling common factors, which can reveal the underlying nature of the locus, such as whether it forms a straight line or another simple geometric shape.

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