Equation of a Locus Equation of a Locus n chapter 1 of this course, methods for analysis of linear equations are presented. If a group of x and y values [or ordered pairs, P(x,y)] that satisfy a given linear equation are plotted on a coordinate system, the resulting graph is a straight line. When higher-ordered equations such as are encountered, the resulting graph is not a straight line. However, the points whose coordinates satisfy most of the equations in x and y are normally not scattered in a random field. If the values are plotted, they will seem to follow a line or curve (or a combination of lines and curves). In many texts the plot of an equation is called a curve, even when it is a straight line. This curve is called the locus of the equation. The locus of an equation is a curve containing those points, and only those points, whose coordinates satisfy the equation. At times the curve may be defined by a set of conditions rather than by an equation, though an equation may be derived from the given conditions. Then the curve in question would be the locus of all points that fit the conditions.
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For instance a circle may be said to be the locus of all points in a plane that is a fixed distance from a fixed point. A straight line may be defined as the locus of all points in a plane equidistant from two fixed points. The method of expressing a set of conditions in analytical form gives an equation. Let us draw up a set of conditions and translate them into an equation. If a point moves according to some fixed rule, its co-ordinates will always satisfy some algebraic relation corresponding to the fixed rule. The resulting path (a curve) of the moving point is called the locus of the point. The locus i.e. the curve now contains all the points satisfying the specified condition and no point outside the curve satisfies the condition. When a point moves in a plane under certain geometrical conditions, the point traces out a path. This path of the moving point is called its locus. Equation of Locus :- The equation to the locus is the relation which exists between the coordinates of all the point on the path, and which holds for no other points except those lying on the path. Procedure for finding the equation of the locus of a point (i) If we are finding the equation of the locus of a point P, assign coordinates (h, k) to P. (ii) Express the given conditions as equations in terms of the known quantities and unknown parameters. (iii) Eliminate the parameters, so that the eliminant contains only h, k and known quantities.
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(iv) Replace h by x, and k by y, in the eliminant. The resulting equation is the equation of the locus of p. The problem of determining the equation of locus of points every pair of which has constant slope. PRACTICE PROBLEMS: Find the equation of the curve that is the locus of all points equidistant from the following: l. The points (0,0) and (5,4). 2. The points (3, - 2) and ( - 3,2). 3. The line x = - 4 and the point (3,4). 4. The point (4,5) and the line y = 5x - 4. HINT: Use the standard distance formula to find the distance from the point P(x,y) and the point P(4,5). Then use the formula for finding the distance from a point to a line, given in chapter 1 of this course, to find the distance from P(x,y) to the given line. Put the equation of the line in the form Ax + By + C=O.
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