PLAN Application of inequality sum and difference, along with lengths of perpendicular. For this type of questions involving inequally we should always check all options.
Situation analysis Check all the inequalities according to options and use lengths of perpendicular from the point `(x_(1)y_(1))` to `ax+by+c=0`
i.e. `(|ax_(1)+by_(1)+c|)/(sqrt(a^(2)+b^(2)))`
As `a gt b gt c gt 0`
`a-c gt 0` and `b gt 0`
`implies a+b-c gt 0`.......`(i)`
`a-b gt 0` and `c gt 0`.....`(ii)`
`:.` Option `(c )` are correct.
Also, the point of intersection for `ax+by+c=0` and `bx+ay+c=0`
i.e. `((-c)/(a+b),(-c)/(a+b))`
The distance between `(1,1)` and `((-c)/(a+b),(-c)/(a+b))`
i.e. less than `2sqrt(2)`,
`impliessqrt((1+(c )/(a+b))^(2)+(1+(c )/(a+b))^(2)) lt 2sqrt(2)`
`implies((a+b+c)/(a+b))sqrt(2) lt 2sqrt(2)`
`impliesa+b+c lt 2a+2b`
`impliesa+b-c gt 0`
From Eqs. `(i)` and `(ii)` , option `(a)` is correct.
