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

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Samyak Jain · Answer

Let length of perpendicular from origin on the line be d and be the angle between the perpendicular and the x-axis. Then x-intercept is d sec and y-intercept is d cosec of the line. It is given that x and y intercepts of the line are a and b respectively. a = d sec & b = d cosec i.e. cos = d/a sin = d/b cos 2 + sin 2 = 1 d 2 / a 2 + d 2 / b 2 = 1 or 1/a 2 + 1/b 2 = 1/d 2 …...(1) Note that perpendicular from origin to the line will not change even if the axes are rotated. Now, let be the angle between line and new x-axis. Similarly, p = d sec & q = d cosec cos = d/p & sin = d/q 1/p 2 + 1/q 2 = 1/d 2 …..(2) From (1) & (2), 1/a 2 + 1/b 2 = 1/p 2 + 1/q 2

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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

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