Home
Class 12
MATHS
If (x , y) and (X ,Y) are the coordinate...

If `(x , y)` and `(X ,Y)` are the coordinates of the same point referred to two sets of rectangular axes with the same origin and it `u x+v y ,` where `u` and `v` are independent of `xa n dy` , becomes `V X+U Y ,` show that `u^2+v^2=U^2+V^2dot`

Text Solution

Verified by Experts

Let the axes rotate at angle `theta`. If `(x,y)` is the point with respect to the old axes and (x,y) are the coordinates with respect to the new axes, then
`{(x,=,Xcostheta-Ysintheta,,),(y,=,Xsintheta+Ycostheta,,):}`
Then `ux+vy=u(Xcostheta-Ysintheta)+v(Xsintheta + Y costheta`
`=(ucostheta +vsintheta )X+(-usintheta +vcos theta )Y`
But given new exression is `VX+UY`. Then,
`VX+UY=(ucostheta +vsintheta )X+(-usintheta +vcos theta) Y`
On comparing the coefficients of X and Y, we get
`ucostheta +vsintheta=V` (1)
and `-usintheta +vcostheta=U` (2)
Squareing and adding (1) and (2) , we get
`u^2+v^2=U^2+V^2`
Promotional Banner

Topper's Solved these Questions

  • COORDINATE SYSYEM

    CENGAGE PUBLICATION|Exercise Concept applications 1.1|6 Videos

  • COORDINATE SYSYEM

    CENGAGE PUBLICATION|Exercise Concept applications 1.2|8 Videos

  • COORDINATE SYSYEM

    CENGAGE PUBLICATION|Exercise Illustration1.67|1 Videos

  • COORDINATE SYSTEM

    CENGAGE PUBLICATION|Exercise Multiple Correct Answers Type|2 Videos

  • CROSS PRODUCTS

    CENGAGE PUBLICATION|Exercise DPP 2.2|13 Videos

Similar Questions

Explore conceptually related problems

For 10 values of X, it is given that sumu = 4 and sum u^2 = 144 , where u = (x-10)/5 , find sumx^2

If (x+i y)^3=u+i v , then show that u/x+v/y=4(x^2-y^2)

A particle starts with the velocity u and moves in a straight line, its acceleration being always equal to its displacement. If v be the velocity when its displacement is x, then show that v^2= u^2+x^2 .

If 2u = 5x and the harmonic mean of x is -0.4, then find the harmonic mean of u.

If 2u = 5x and the harmonic mean of x is 0.4, then find the harmonic mean of u.

A particle starts with the veloctiy u and moves in a straight line, its acceleration being always equal to its displacement. If v be the velocity when its displacement is x, the show that v^(2)=u^(2)+x^(2) .

If a variate x is expressed as a linear function of two variates u and v in the form x=au+bv , then mean bar(x) of x is

Choose the correct alternative : [MCQ] If x and y are two independent variables then V(x - y) is equal to—

If a ,b ,c and u ,v ,w are the complex numbers representing the vertices of two triangles such that (c=(1-r)a+r b and w=(1-r)u+r v , where r is a complex number, then the two triangles (a)have the same area (b) are similar (c)are congruent (d) None of these