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Line L has intercepts a and b on the coo...

Line `L` has intercepts `a and b` on the coordinate axes. When the axes are rotated through a given angle keeping the origin fixed, the same line `L` has intercepts `p and q`. Then (a) `a^2+b^2=p^2+q^2` (b) `1/(a^2)+1/(b^2)=1/(p^2)+1/(q^2)` (c) `a^2+p^2=b^2+q^2` (d) `1/(a^2)+1/(p^2)=1/(b^2)+1/(q^2)`

A

(a) `a^2+b^2=p^2+q^2`

B

(b) `1/(a^2)+1/(b^2)=1/(p^2)+1/(q^2)`

C

(c) `a^2+p^2=b^2+q^2`

D

(d) `1/(a^2)+1/(p^2)=1/(b^2)+1/(q^2)`

Text Solution

Verified by Experts


As shown in the figure, axes are rotated by an angle `theta` about origin in anticlockwise direction.
Line L has intercepts a and b on original axes and intercepts p and q on rotated axes.
Equation of line L w.r.t. original axes is
`(x)/(a) + (y)/(b)-1= 0 " " (1)`
Equation of line L w.r.t. rotated axes is
`(x)/(p) + (y)/(q)-1= 0 " " (2)`
Since origin and line L are not changing their positions, distance of line from origin remains the same even though the axes are rotated.
So, comaring distances of lines given by equations (1) and (2) from origin, we get
`(|(0)/(a) + (0)/(b)-1|)/(sqrt((1)/(a^(2))+ (1)/(b^(2)))) = (|(0)/(p) + (0)/(q)-1|)/(sqrt((1)/(p^(2))+ (1)/(q^(2))))`
`rArr (1)/(a^(2)) + (1)/(b^(2)) = (1)/(p^(2)) + (1)/(q^(2))`
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