Question

Find the middle point of the chord intercepted on line `lx + my + n = 0` by circle `x^2+y^2=a^2`.
A. `((-l)/(l^(2)+m^(2)),(-m)/(l^(2)+m^(2)))`
B. `((-ln)/(l^(2)+m^(2)),(-mn)/(l^(2)+m^(2)))`
C. `((-l)/(n(l^(2)+m^(2))),(-m)/(n(l^(2)+m^(2))))`
D. none of these

Answer 1

Correct Answer - B
Let `(x_(1),y_(1))` be the mid-point of the chord intercepted by the circle `x^(2)+y^(2)=a^(2)` on the line `lx+my+n=0`.
Then, the equation of the chord of the chord of the circle `x^(2)+y^(2)=a^(2)`
whose middle point is `(x_(1), y_(1))` is
`x x_(1) + y y_(1)-a^(2)=x_(1)^(2)+y_(1)^(2)-a^(2)`
`rArr x x_(1)+y y_(1)=x_(1)^(2)+y_(1)^(2) " " ...(i)`
Clearly, lx+my+n=0 and (i) represents the same line.
`:. (x_(1))/(l)=(y_(1))/(m)=(-(x_(1)^(2)+y_(1)^(2)))/(n)=lambda`, say
`rArr x_(1)=l lambda, y_(1)=m lambda` and `x_(1)^(2)+y_(1)^(2)=-n lambda`
`rArr (l^(2)+m^(2))lambda^(2)=-n lambda`
`rArr lambda=-(n)/(l^ (2)+m^(2))`
`:. x_(1)=-(ln)/(l^(2)+m^(2)), y_(1)=(-mn)/(l^(2)+m^(2))`
Hence, the required point is `((-ln)/(l^(2)+m^(2)),(-mn)/(l^(2)+m^(2)))`