Question

Explain - What is locus of a point whose coordinates are given? With examples and types of problems related to it ..

Topic -: Coordinate Geometry .

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Answer 1

Explanation:-

Locus:-

When a point moves 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 a locus is the relation which exists between the coordinates of any point on the path and

Which holds for no other poin except those lying on the path . in other words equation to a curve ( or locus) is

The equation connecting the x and the y coordinates of every point on he curve

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 coordinate ( h , k ) to p

ii) Express the given condition in term of the known quantities to facilitate calculation. we sometime include some unknown quantities known as parameters if necessary to relate the given condition

iii) Eliminate the parameters so that the eliminat contains only h , k and know quantities. if h and k coordinate of the moving point are obtained in terms of a third variable 't' called the parameter . Eliminate 't' to obtain the relation in h and k and simplify this relation

iv) Replace h by x and k by y , in the eliminant. The resulting equation would be the equation of the locus of P

Now take two illustration to understand locus

question 1:- Find the locus of a variable point which is at a distance of 2 units from the y - axis

Solution:-

Let p(x,y) be a variable points . its distance from y axis is |x|

therefore | x | = 2 or x = +2 and - 2

Question 2:- Find the locus of a variable point whose distance from A( 4 , 0) is equal to its distance from B ( 0, 2 )

Solution:-

Let the coordinate of P be ( x , y )

AP² = ( x - 4 )² + ( y - 0 )² = x² + y² - 8x + 16 and

BP² = ( x - 0 )² + ( y - 2 )² = x² + y² - 4y + 4

Since AP² = BP²

=> x² + y² - 8x + 16 = x² + y² - 4y + 4

=> 8x - 4y - 12 = 0 or 2x - y - 3 = 0