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Co-ordinate Geometry

The curve described by a point which moves under given condition(s) is called its locus.
In other words a locus is the set of points(and only those points) that satisfy the given geometric condition(s).
It follows that:
(i) Every point satisfying the given conditions is a point on the locus
(ii) Every point on the locus satisfies the given conditions.

Equation to the locus of a point

The equation to the locus of a point is the relation which is satisfied by the coordinates of every point on the locus of the point.

In order to find the locus of a point, the following algorithm can be used:

Step I: Assume the coordinates of the point say, (h, k) whose locus is to be found.
Step II: Write the given condition in mathematical form involving h, k.
Step III: Eliminate the variable(s), if any
Step IV: Replace h by x and k by y in the result obtained in step III.
The equation so obtained is the locus of the point which moves under some stated condition(s).

Example:

Find the locus of set of points in a plane which are equidistant from two given points A and B.

Sol: The locus of a point equidistant from two given points A and B is represented as {P/ AP = BP} i.e., the set of points P such that AP = BP. This set represents the perpendicular bisector of the line segment AB which is the locus. Any point on it is equidistant from the two points A and B.

Parametric equation of a locus

Sometimes x and y coordinates of a point of the locus obeying a property may be given as separate function of a variable.
Say x = f(t) and y = g(t).
Now for all allowed values of t ∈ R, the point (x, y) defines a locus.
Hence, the relation between x and y is indirectly controlled by the variable t.
The variable t is called the parameter and x = f(t) and y = g(t) are called the parametric equations of the locus.

Example Find the locus of the point represented by x = t2 + t + 1, y = t2 – t + 1.
Sol:
x + y = 2(t2 + 1) and x – y = 2t
From the latter equation, t = (x – y)/2
Substituting in the former equation, we have
⇒ x + y = 2[(x – y)/2)2 + 1]
⇒ 2x + 2y = x2 – 2xy + y2 + 4
∴ x2 – 2xy + y2 – 2x – 2y + 4 = 0

Cartesian equation of a locus from parametric equations

The Cartesian equation of a locus is a direct relation between x and y.
The parameter 't' is eliminated from the two relations to obtain a functional relation between x and y.

Some times the locus of a point P, according to a given geometrical condition, could give a region.
Ex: Locus of a point P(x, y) such that | x | < 1, | y | < 2 constitute a region inside rectangle.

| x | < 1 ⇒ – 1 < x < 1; | y | < 2 ⇒ – 2 < y < 2
In order to find the region of | x | < 1 and | y | < 2, first we will draw the lines of x = – 1, x = 1 and y = – 2, y = 2 in the cartesian plane.

Clearly, the region represented by | x | < 1 and | y |< 2 is a inside space of the rectangle formed by the lines x = – 1, x = 1 and y = – 2, y = 2.

Transformation of axes

Reflection

The concept of reflection is introduced first although transformation of axes is not involved in it.

Pre-image: The original shape of an object is called the pre-image.

Reflection: A reflection is a kind of transformation which is a 'flip' of a shape over a line (or a point or a plane) of reflection. It is always the mirror image of an object i.e, with right and left reversed. An object and its reflection have the same shape and size, but the figures face in opposite directions. A reflection can be seen, for example, in water, in mirror, or in shiny surface. See adjacent figures.

Image: The reflection of an object (pre-image) is called the image of that object.

Line reflection: A reflection over a line is a transformation in which each point of the original figure has an image that is at the same distance from the line of reflection as the original point but on the opposite side of the line. When a figure is reflected over a line the image is congruent to its pre-image and this property is known as isometry.

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