LOCUS

1) A point moves in the xy-plane in such a way that its distance from the point (4,0) is always equal to its distance from the y-axis. Find the equation to the locus of the moving point.                                  y²=8(x- 2)

2) A point moves in the xy-plane in such a way that its distance from the point (a,0) is always greater than the abscissa of the point by a. Find the equation to the locus of the moving point.                 y²= 4ax

3) Find the equation to the locus of a point if Its distance from the x-axis is double its distance from the point (1,1).    4x²+ 3y²- 8x - 8y+ 8= 0

4) The distance of a moving from x-axis is twice the distance of the point from the point (1,1). Find the equation to the locus of the point.         4x²+ 3y² - 8x - 8y +8= 0

5) A point moves in a plane such that its distance from the point (2,3) exceeds its distance from the y-axis by 2. Find the equation of the locus of the point.                      (y- 3)² = 8x.

6) Find the equation of the locus of a moving point which is always equidistant from the points (-1,1) and (3,-2).                         8x - 6y= 11.

7) Find the equation of the locus of a moving point which is always equidistant from the points (a+ b, a- b) and (a- b, a+b).                         y= x

8) Find the equation of the locus of a point which always forms an isosceles triangle with the points (5,2) and (-3,6).                 y= 2(x+1).

9) A and B are two fixed points whose coordinates are (2,1) and (3,2) respectively. A point P moves in such a way that PA= 2PB always. Find the locus of P.            3x² + 3y²- 20x - 14y+47= 0

10) The coordinates of two points A and B are (-5,3) and (2,4) respectively. Find the locus of P(x,y) such that PA:PB= 3:2.         5x²+ 5y² - 76x - 48y+ 44= 0

11) Find the locus of the point which moves such that its distance from the points (3,1) and (-2, 5) are always equal.            10x- 8y +19= 0

12) Find the locus of a point which moves so that it is always at a distance 4 from (1,-2).       x²+ y² - 2x + 4y -11= 0

13) Find the locus of a point which moves so that its distance from (4,0) is always twice its distance from (1,0).                           x²+ y²= 4

14) P is such a variable point that the sum of the squares of its distances from the points (-2,0) and (2,0) is always equal to 40. Find the equation of the locus of P.           x²+ y²= 16

15) Find the locus of a moving point so that its distance from (0,4) is two-thirds of its distance from (0,9).                                x²+ y²= 36

16) A points moves so that its distance from the point (a,0) exceeds the distances from axis of y by a. find the locus.             y²= 4ax

17) Find the locus of a point which moves so that its distance from the y-axis is double its distance from the point (2,2).           3x²+ 4y² - 16x - 16y +32= 0

18) Find the locus of a point which moves such that its distance from the point (4,0) is √2 times of its distance from the y-axis.      x² - y² + 8x - 16= 0

19) Find the locus of a point which moves so that its distance from the point (0,5) is two thirds of its distance from the x-axis.           9x²+ 5y² - 90y + 225= 0

20) The points A and B are (-4,0) and (-1,0) respectively. A point P moves in such a way that PA: PB=2:1. find the locus of P.          x²+y²= 4

21) Find the locus of a point which moves so that the sum of squares of its distances from the two points (3,0) and (-3,0) is 36.       x²+ y²= 9

22) Find the locus of a point which moves so that the sum of the squares of its distances from the points (3,0) and (-3,0) is always equal to 50.                      x²+ y²= 16

23) Find the locus of a point which moves so that the sum of its distances from the points (4,0) and (-4,0) is 10.               9x²+ 25y²= 225

24) Find the locus of a point which moves such that the difference of its distances from (2,5) and (6,5) is 2.                 3x²-y²- 24x+ 10y +20= 0

25) Find the locus of the centre of the circles passing through the point (c,0) and (- c ,0).                x= 0

26) A(2,-3) and B(4,0) are two points. Find the locus of a point P such that the area of the triangle PAB is always for units.      3x- 2y= 4

27) A line segment, 16 units in length, moves so that its ends are always on the positive coordinate axes. Find the equation of the locus of its mid-point.                 x²+y²= 64

28) If the coordinates of two vertices of a triangle be A(-3,0) and B(3,0) and angle ACB=90°, Find the locus of the centroid of the triangle.                              x²+ y²= 11

29) The coordinates of a moving point P are (a sec t, b tan t), where t is a variable parameter. Find the locus of the point P.     x²/a² - y²/b² = 1

30) The coordinates of a moving point P are {(t-1)/(t+1), (2t+1)/(t+1)}, where t is a variable parameter. Find the locus of the point P.                               x- 2y+3= 0

31) AB is a line of fixed length, 6 units, joining the points A(t,0) and B which lies on the positive y-axis. P is a point on AB distance 2 units from A. Express the coordinates of B and of P in terms of t. Find the locus of P as t varies.           x²+ 4y²= 16

32) Show that the locus of the point of intersection of the straight lines x sin t - y(cos t -1)= a sin t; x sin t - y(cos t + 1) = - a sin t is x² + y² = a².

33) A(1,2) and B(3,4) are two fixed points; P(x,y) divides AB internally in the ratio 1: m. Find the coordinates of P. If m is a variable, deduce from the above, equation of the straight line joining A and B.            {(m+3)/(m+1), (2m+4)/(m+1)}; x - y+1= 0

**34) A point moves in such a way that the sum of its distances from two fixed points (ae,0) and (-ae,0) is 2a, prove that the equation of its locus is x²/a² + y²/b² = 1 where b²= a²(1- e²).