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

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.