Question

The number of lines that are parallel to \[2x + 6y + 7 = 0\;\]and have an intercept of length \[10\] between the coordinate axes is
\[\begin{array}{*{20}{l}}{A)\;\;\;\;\;\;\;\;\;\;\;{\rm{ }}1\;\;\;\;}\\{B)\;\;\;\;\;\;\;\;\;\;\;{\rm{ }}2}\\{C)\;\;\;\;\;\;\;\;\;\;\;{\rm{ }}4\;\;\;\;}\\{D)\;\;\;\;\;\;\;\;\;\;\;{\rm{ }}Infinitely{\rm{ }}many}\end{array}\]

Answer 1

Hint:Straight line is a set of infinite points in which all points are linear. All parallel lines have equal slopes. The intercept between the axes means the distance between the intercept on the x and y axes. The distance formula is used to determine the distance between two points.

Formula used:
Slope of two parallel lines are equal
\[{m_1} = {m_2}\]
Where
\[{m_1}\] Is a slope of given line
\[{m_2}\]Is a slope of lines which are parallel to given line
Equation of straight line:
\[y = mx + c\]
Where
m is the slope of the line
c is y-intercept of the line
Distance formula:
\[d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \]

Complete step by step Solution:
Given:
Equation of line, the y-intercept of the line.
\[2x + 6y + 7 = 0\;\]
The above equation can be written as
\[y = \dfrac{{ - 2x - 7}}{6}\]
\[y = - \dfrac{1}{3}x - \dfrac{7}{6}\]
Slope of given line is \[ - \dfrac{1}{3}\]
Slope of all lines parallel to given lines is also \[ - \dfrac{1}{3}\]
Now equations of the required lines are:
\[y = mx + c\]
\[y = - \dfrac{1}{3}x + c\]
Put y equal to zero in order to find x intercept
X intercept is \[(3c,0)\]
Put x equal to zero in order to find the y-intercept
Y intercept is \[(0,c)\].
Now use the distance formula in order to find the intercept
\[d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \]
Intercept between the coordinate axes is given in question which is equal to \[10\]
\[10 = \sqrt {{{\left( {3c} \right)}^2} + {{\left( c \right)}^2}} \]
\[100 = {\left( {3c} \right)^2} + {\left( c \right)^2}\]
\[100 = 10{\left( c \right)^2}\]
\[{c^2} = 10\]
\[c = \pm \sqrt {10} \]
Now equations of lines are
\[y = - \dfrac{1}{3}x + \sqrt {10} \] and \[y = - \dfrac{1}{3}x - \sqrt {10} \]
There are two parallel lines parallel to the given lines.

Therefore, the correct option is (B).

Note: Line which are parallel to each other are having equal slope. Every line makes some angle with axes. Slope of a line is also equal to the tan of angle which lines make with axes. Slope is also known as gradient.
Distance formula is used to calculate the distance between x and y intercept.