FIND THE LOCUS OF THE MOVING POINT P

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 :

(1) + (2)  

cos²θ + sin²θ  =  (x/a)^2/3 + (y/a)^2/3

1  =  (x^2/3 + y^2/3)/a^2/3

a^2/3  =  x^2/3 + y^2/3

x^2/3 + y^2/3  =  a^2/3

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

AB²  =  OA² + OB²

8²  =  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

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