Question

What is the distance between the lines 4x + 3y = 11 and 8x + 6y = 15?
(a). \[\dfrac{7}{2}\]
(b). 4
(c). \[\dfrac{7}{{10}}\]
(d). None of these

Answer 1

Hint: Check if the two lines are parallel using the condition \[\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}}\]. Then use the distance between two parallel lines formula for the lines \[ax + by + {c_1} = 0\] and \[ax + by + {c_2} = 0\] is given as \[d = \dfrac{{|{c_1} - {c_2}|}}{{\sqrt {{a^2} + {b^2}} }}\] to find the answer.

Complete step-by-step answer:

We are given the equations of the two lines as 4x + 3y = 11 and 8x + 6y = 15.

The condition for two lines \[{a_1}x + {b_1}y + {c_1} = 0\] and \[{a_2}x + {b_2}y + {c_2} = 0\] to be parallel is given by:

\[\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}}\]

Let us check this condition for the lines 4x + 3y = 11 and 8x + 6y = 15.

\[\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{4}{8}\]

\[\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{1}{2}..............(1)\]

\[\dfrac{{{b_1}}}{{{b_2}}} = \dfrac{3}{6}\]

\[\dfrac{{{b_1}}}{{{b_2}}} = \dfrac{1}{2}............(2)\]

From equations (1) and (2), we have:

\[\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}}\]

Hence, the two lines are parallel.

For finding the distance between the two parallel lines, we first express the two equations such that the coefficients of x and y are equal.

We multiply the equation 4x + 3y = 11 by 2, then, we have:

\[2(4x + 3y) = 2(11)\]

Simplifying, we have:

\[8x + 6y = 22\]

Now, we use the formula for calculating the distance between two parallel lines \[ax + by + {c_1} = 0\] and \[ax + by + {c_2} = 0\] given as follows:

\[d = \dfrac{{|{c_1} - {c_2}|}}{{\sqrt {{a^2} + {b^2}} }}\]

From the equations of the lines, we have:

\[{c_1} = - 22\]

\[{c_2} = - 15\]

a = 8

b = 6

Then, we have:

\[d = \dfrac{{| - 22 - ( - 15)|}}{{\sqrt {{8^2} + {6^2}} }}\]

Simplifying, we have:

\[d = \dfrac{{| - 22 + 15|}}{{\sqrt {64 + 36} }}\]

\[d = \dfrac{{| - 7|}}{{\sqrt {100} }}\]

\[d = \dfrac{7}{{10}}\]

Hence, the correct answer is option (c).

 

Note: You may forget to convert the equations such that both equations have same coefficients of x and y, in that case, you get a wrong answer. Always convert both equations into similar form.