(i) If a, b, c, are in A.P. then show that `ax….+….by +c=0` passes through a fixed point. Find then fixed point.
(ii) If `9a^(2)+16b^(2)-24ab-25c^(2)=0,` then the family of straight lines `ax+by+0` is concurrent at the point whose co-ordinates are given by `"________"`
(iii) If `3a+4b-5c=0,` then the family of straight lines `ax+by+c=0` passes through a fixed point. Find the coordinates of the point.

Explanation

(i) a,b,c are in A.P.
`impliesa-2b+c=0`
Comparing the given equation `ax+by+c=0` with `a-2b+c=0` we find that the straight line `ax+by6+c=0` passes through the fixed point `(1,-2).`
(ii) We have,
`9a^(2)+16b^(2)-24ab-25c^(2)=0`
`implies(3a-4b)^(2)-(5c)^(2)=0`
`implies(3a-4b+5c)(3a-4b-5c)=0`
`implies((3)/(5)a-(4)/(5)b+c)((-3)/(5)a+(4)/(5)b+c)=0`
`implies` The family of lines `ax+by+c=0` is either concurrent at `((3)/(5),(-4)/(5))or (-(3)/(5),(4)/(5))`
(iii) We have,
`3a+4b-5c=0`
`implies-(3)/(5)a-(4)/(5)b+c=0`
`implies` The given family of lines `ax+by+x=0` passes through the fixed point `((-3)/(5),(-4)/(5)).`