Assume P(xp,yp), A(xa,ya), etc. We know that rotation rule of 90<span>° clockwise about the origin is R_-90(x,y) -> (y,-x) For example, rotating A about the origin 90</span><span>° clockwise is (xa,ya) -> (ya, -xa) or for a point at H(5,2), after rotation, H'(2,-5), etc.
To rotate about P, we need to translate the point to the origin, rotate, then translate back. The rule for translation is T_(dx,dy) (x,y) -> (x+dx, y+dy)
So with the translation set at the coordinates of P, and combining the rotation with the translations, the complete rule is: T_(xp,yp) R_(-90) T_(-xp,-yp) (x,y) -> </span>T_(xp,yp) R_(-90) (x-xp, y-yp) -> T_(xp,yp) (y-yp, -(x-xp)) -> (y-yp+xp, -x+xp+yp)
Example: rotate point A(7,3) about point P(4,2) => x=7, y=3, xp=4, yp=2 => A'(3-2+4, -7+4+2) => A'(5,-1)
<span>For given hyperbola: center: (0,0) a=7 (distance from center to vertices) a^2=49 c=9 (distance from center to vertices) c^2=81 c^2=a^2+b^2 b^2=c^2-a^2=81-49=32 Equation of given hyperbola:
.. 2: vertices (0,+/-3) foci (0,+/-6) hyperbola has a vertical transverse axis Its standard form of equation: , (h,k)=(x,y) coordinates of center For given hyperbola: center: (0,0) a=3 (distance from center to vertices) a^2=9 c=6 (distance from center to vertices) c^2=36 a^2+b^2 b^2=c^2-a^2=36-9=25 Equation of given hyperbola: </span>
