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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

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

Grade:11

1 Answers

Samyak Jain
333 Points
2 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  \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 \beta be the angle between line and new x-axis.
Similarly, p = d sec\beta  &  q = d cosec\beta 
cos\beta = d/p  &  sin\beta = d/q
1/p2 + 1/q2 = 1/d2     …..(2)
From (1) & (2),
1/a2 + 1/b2  =  1/p2 + 1/q2 

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