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Basic Concepts Of Straight line Part 2

Posted on - 04-01-2017

Math

IIT JEE

Locus

When a point moves in a plane under certain geometrical conditions, the point traces out a path. This path of a moving point is called its locus.

Equation of Locus:

The equation to a locus is the relation which exists between the coordinates of any point on the path, and which holds for no other point except those lying on the path.

Procedure for finding the equation of the locus of a point

  1. If we are finding the equation of the locus of a point P, assign coordinates (h, k) to P.
  2. Express the given conditions as equations in terms of the known quantities to facilitate calculations, We sometimes include some unknown quantities known as parameters.
  3. Eliminate the parameters, so that the eliminant contains only h, k and known quantities.
  4. Replace h by x, and k by y, in the eliminant. The resulting equation would be the equation of the locus of P.
  5. If x and y coordinates of the moving point are obtained in terms of a third variable t (called the parameter), eliminate t to obtain the relation in x and y and simplify this relation. This will give the required equation of locus.

Example 1

The ends of a rod of length move on two mutually perpendicular lines. Find the locus of the point on the rod, which divides it in the ratio 2 : 1.

Solution:

Suppose the two perpendicular lines are x = 0 and y = 0 and the rod intercepts a and b cuts on these two lines respectively, then the two points on these lines are (0, a) and (b, 0). The point P has coordinates given by h = , k =

Also = a2 + b2

Thus the required locus is x2+.

Example 2

Find the locus of a point which moves so that the sum of its distances from (3, 0) and (–3, 0) is less than 9.

Solution:

Let P(h, k)be the moving point such that the sum of its distances from A (3, 0) and B (–3, 0) is less than 9. .

Then, PA + PB < 9


&⇒


&⇒


&⇒
(h-3)2 + k2 < 81 + (h+3)2 + k2 – 18


&⇒
–12h – 81 < –18


&⇒
4h + 27 > 6


&⇒
(4h + 27)2 > 36 [(h+3)2 + k2]


&⇒
20h2 + 36k2 < 405

Hence the locus of (h, k) is

20x2 + 36y2 < 405.

 
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