# A line is making intercept a and b on coordinate axis. its axis are rotated by an angle without shifting its origin and due to this intercept changed to p and q. Show that 1/a2 +1/b2 = 1/p2 +1/q2

Samyak Jain
333 Points
4 years ago
Let length of perpendicular from origin on the line be d and
$\alpha$ be the angle between the perpendicular and the x-axis.
Then x-intercept is d sec$\alpha$ and y-intercept is d cosec$\alpha$ of the line.
It is given that x and y intercepts of the line are a and b respectively.
$\therefore$ a = d sec$\alpha$  &  b = d cosec$\alpha$
i.e. cos$\alpha$ = d/a  sin$\alpha$ = d/b
cos2$\alpha$ + sin2$\alpha$ = 1  $\dpi{100} \Rightarrow$  d2 / a2 + d/ b2 = 1  or  1/a2 + 1/b2 = 1/d2   …...(1)
Note that perpendicular from origin to the line will not change even if the axes are rotated.
Now, let $\dpi{100} \beta$ be the angle between line and new x-axis.
Similarly, p = d sec$\dpi{100} \beta$  &  q = d cosec$\dpi{100} \beta$
cos$\dpi{100} \beta$ = d/p  &  sin$\dpi{100} \beta$ = d/q
1/p2 + 1/q2 = 1/d2     …..(2)
From (1) & (2),
1/a2 + 1/b2  =  1/p2 + 1/q2