Question

Find the equations of the lines, which cut-off intercepts on the axes whose sum and product are 1 and –6, respectively.

Answer 1

The sum of cut-off intercepts is 1 and the product is −6 respectively.

Let the intercepts cut by the given lines on the axes be a,b respectively.

According to the question,

a+b=1 a⋅b=−6

Further solve the above two equations.

If b= −6 a then,

a+ −6 a =1 a 2 −6 a =1 a 2 −6=a a 2 −a−6=0

Further simplify the above expression.

a 2 −3a+2a−6=0 a( a−3 )+2( a−3 )=0 ( a+2 )( a−3 )=0 a=−2 or a=3

If a=−2 ⇒ b= −6 −2 =3(1)

If a=3 ⇒ b= −6 3 =−2(2)

The formula for the equation of the line whose intercepts on the axes are a and b respectively is given by,

x a + y b =1 bx+ay−ab=0 (3)

If the values of equation (1) have been used as intercepts in equation (3).

−2x+3y+6=0.

If the values of equation (2) have been used as intercepts in equation (3).

3x−2y+6=0

Thus, the equations of lines whose cut-off intercepts on the axes with sum and product of 1 and −6 respectively are −2x+3y+6=0 and 3x−2y+6=0.