Question

# The line L has intercepts a and b on the coordinate axes. When keeping the origin fixed, the coordinate axes are rotated through a fixed angle, then the same line has intercepts p and q on the rotated axes. Then prove that1a2+1b2=1p2+1q2

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Solution

## Suppose the axes are rotated in the anti-clockwise direction through an angle α. The equation of the line L with respect to the old axes is given by x/a+y/b=1. To find the equation of L with respect to the new axes, we replace x by xcosα−ysinα and y by xsinα+ycosα, so that the equation of L with respect to the new axes is1a(xcosα−ysinα)+1b(xsinα+ycosα)=1Since p,q are the intercepts made by this line on the coordinate axes, we have on putting (p,0) and then (0,q)1/p=(1/a)cosα+(1/b)sinαand 1/q=−(1/a)sinα+(1/b)cosαEliminate αSquaring and adding we get1/p2+1/q2=1/a2+1/b2

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