Question

when the axes of rotated through 90⁰ find the transformed equation x²/a²-y²/b² =1 find the answer ​

Answer 1

Answer:

$\frac{Y²}{a²} - \frac{X²}{b²} = 1$

EXPLAINATION

Let

(x,y) = coordinates of a point in the original axes

(X, Y) = the co-ordinate of the same point in the new rotated axes

GIVEN EQUATION

$\frac{x²}{a²} - \frac{y²}{b²} = 1-----(1)$

RELATION OF (x,y) and (X, Y)

Rotating axes by 90 is similar to rotating (x,y) by - 90 about origin. One way to do this is multiplying

(x + iy) by $exp(-i\frac{π}{2})$.

$X + iY = (x + iy)exp(-i\frac{π}{2})$

$(X + iY)exp(i\frac{π}{2}) = (x + iy)$

$(x + iy) = (X + iY)[ cos(\frac{π}{2} )+ isin(\frac{π}{2} )]$

$(x + iy) = (X + iY)( i )$

$(x + iy) = (iX + i²Y)$

$(x + iy) = (iX - Y)$

$(x + iy) = ( - Y + iX)$

$(x = - Y,y = X)------(2)$

This relation can also be calculated using transformation matrix.

EQUATION IN NEW CO-ORDINATE

Substituting value of x, y in (1)

$\frac{(-Y)²}{a²} - \frac{(X)²}{b²} = 1$

$\frac{Y²}{a²} - \frac{X²}{b²} = 1$

This is the required equation.