Question

Explanation needed
1) Find the coordinates of the points symmetric with respect to X-axis, to the following points. Plot the points. (a) A(2,3) ; (b) B(-3,2) ; (c) C(-1,-1) ; (d) D(-3,4)

2) Extremities of the diagonal of a parallelogram are the points (1,2) and (5,4). Third vertex is the points (2,3). Then the fourth vertex is
{answer should come (4,3)}

3) Find the distance between two points whose polar co-ordinates are $(2, \frac{ \pi }{3}) and (4, \frac{ \pi }{2}).$ {answer should come$( \sqrt{20-8 \sqrt{3} } )$}

Answer 1

symmetric wrt x axis means, the y coordinate changes sign.
a) A' = (2,-3)    B'  =(-3,-2)  C' = (-1, 1)

A= (1,2)  B=(2,3)  c=(5,4)  D(x,y)
slope AB = (3-2)/(2-1) = 1
slope BC = (4-3)/(5-2)  = 1/3
Slope CD = (y-4)/(x-5) = 1  =>  y + 1 = x
Slope DA = (y-2)/(x-1) = 1/3  =>  3 y - 5 = x
 solving  x = 4  y = 3

distance² = √[ (4 Cos π/2 - 2 cos π/3)² + ( 4 sin π/2 - 2 cos π/3)² ]
         = √ [ 1 + (4-√3)² ] = √[ 20 - 8√3 ]