FIND THE LOCUS OF THE MOVING POINT P

About "Find the Locus of the Moving Point p"

Find the Locus of the Moving Point p :

Here we are going to see how to find the locus of the moving point P.

Find the Locus of the Moving Point p - Practice questions

Question 1 :

If θ is a parameter, find the equation of the locus of a moving point, whose coordinates are

x = a cos3 θ, y = a sin3 θ.

Solution :

x = a cos3 θ

x/a  =  cos3 θ

(x/a)1/3  =  cos θ

Squares on both sides

cos2 θ  =  (x/a)2/3  ---(1)  

y = a sin3 θ

y/a  =  sin3 θ

(y/a)1/3  =  sin θ

Squares on both sides

sin2 θ  =  (y/a)2/3  ---(2) 

(1) + (2)  

cos2θ + sin2θ  =  (x/a)2/3 + (y/a)2/3

1  =  (x2/3 + y2/3)/a2/3

a2/3  =  x2/3 + y2/3

x2/3 + y2/3  =  a2/3

Question 2 :

Find the value of k and b, if the points P(−3, 1) and Q(2,b) lie on the locus of x2 − 5x + ky = 0.

Solution :

The point P lies on the locus P (−3, 1) 

(-3)2 − 5 (-3) + k (1) = 0

9 + 15 + k  =  0

24 + k  =  0

k  =  -24

The point Q lies on the locus Q (2,b)

22 − 5 (2) + k (b) = 0

4 - 10 + (-24) b  =  0

-6 - 24 k  =  0

-24 k  =  6

k  =  -6/24  =  -1/4

Question 3 :

A straight rod of length 8 units slides with its ends A and B always on the x and y axes respectively. Find the locus of the mid point of the line segment AB

Solution :

First let us find midpoint of the line segment AB,

Midpoint  =  (x1 + x2)/2 , (y1 + y2)/2

  =  (a + 0)/2 , (0 + b)/2

  =  a/2, b/2

h  =  a/2     k = b/2

a  =  2h and b = 2k

AB =  OA2 + OB2

8 =  a2 + b2

64  =  (2h)2 + (2k)2

64  =  4h2 + 4k2 

16  =  h2 + k2

By replacing h = x and k = y, we get

x2 + y2  =  16

After having gone through the stuff given above, we hope that the students would have understood "Find the Locus of the Moving Point p". 

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