The equation of line belonging to the family of lines p(2x+3y−13)+q(x−y+1)=0 ,  and which is at maximum distance from the orioin is

Given family of lines p(2x+3y−13)+q(x−y+1)=0⇒2x+3y−13+qp(x−y+1)=0⇒2x+3y−13+λ(x−y+1)=0∵ Let pq=λ⇒x(2+λ)+y(3−λ)+λ−13=0………...(1) The distance from origin to (1) is D=|λ−13|(2+λ)2+(3−λ)2=|λ−13|2λ2−2λ+13 Differentiate with respect to ′λ′⇒dDdλ=2λ2−2λ+13−(λ−13)(4λ−2)22λ2−2λ+132λ2−2λ+13=2λ2−2λ+13−(λ−13)(2λ−1)2λ2−2λ+1332For maximum value of D ,dDdλ=0⇒2λ2−2λ+13=(λ−13)(2λ−1)⇒λ=0 Substituting λ=0 in equation (1) we get 2x+3y−13=0