If the slope of one of the lines represented by `ax^(2)+2hxy+by^(2)=0` is the square of the other , then `(a+b)/(h)+(8h^(2))/(ab)=`

Options

A. 2
B. 4
C. 6
D. 1

Answer

C

Explanation

Let m and `m^(2)` be the slopes of the lines represented by `ax^(2)+2hxy+by^(2)=0`. Then,
`m+m^(2)=-(2h)/(b) andm.m_(2)=(a)/(b)" "[{:(becausem_(1)+m_(2)=-(2h)/(b)),(and m_(1)m_(2)=(a)/(b)):}]`
`rArr" "m+m^(2)=-(2h)/(b)and m^(3)=(a)/(b)`
`rArr" "m+m^(2)=-(2h)/(b) and m =((a)/(b))^(1//3)`
`rArr" "((a)/(b))^(1//3)+((a)/(b))^(2//3)=-(2h)/(b)`
`rArr" "{((a)/(b))^(1//3)+((a)/(b))^(2//3)}^(3)=(-(2h)/(b))^(2)`
`rArr" "(a)/(b)+((a)/(b))^(2)+3((a)/(b))^(1//3)((a)/(b))^(1//3)((a)/(b))^(2//3){((a)/(b))^(1//3)+((a)/(b))^(2//3)}=-(8h^(3))/(b^(3))`
`rArr" "(a)/(b)+(a^(2))/(b^(2))+2((a)/(b))(-(2h)/(b))=-(8h^(3))/(b^(3))[because((a)/(b))^(1//3)+((a)/(b))^(2//3)=-(2h)/(b)]`
`rArr" "(a)/(b)+(a^(2))/(b^(2))+(8g^(3))/(b^(3))=(6ah)/(b^(2))`
`rArr" "(a(a+b))/(b^(2))+(8h^(3))/(b^(3))=(6ah)/(b^(2))`
`rArr" "(a+b)/(h)+(8h^(2))/(ab)=6" "[-"Multiplying both sides by"(b^(2))/(ah)]`