Question

The perpendicular distance of a straight line \[12x + 5y = 7\] from the origin is given by
A.\[\dfrac{7}{{13}}\]
B.\[\dfrac{{12}}{{13}}\]
C.\[\dfrac{5}{{13}}\]
D.\[\dfrac{1}{{13}}\]

Answer 1

Hint: Given is the equation of a line of the form \[ax + by + c = 0\]. The perpendicular distance can be calculated with the help of a formula that calculates the distance between straight lines from the origin. We will use that formula.

Complete step-by-step answer:
Given the equation of line is,
\[12x + 5y = 7\]
It is of the form,
\[ax + by + c = 0\]
We know that the perpendicular distance is given by the formula,
\[\left| d \right| = \left| {\dfrac{c}{{\sqrt {{a^2} + {b^2}} }}} \right|\]
So on comparing the given equation with the standard equation we get a=12, b=5 and c=-7.
Putting these values in the formula,
\[\left| d \right| = \left| {\dfrac{{ - 7}}{{\sqrt {{{12}^2} + {5^2}} }}} \right|\]
Taking the squares,
\[\left| d \right| = \left| {\dfrac{{ - 7}}{{\sqrt {144 + 25} }}} \right|\]
\[\left| d \right| = \left| {\dfrac{{ - 7}}{{\sqrt {169} }}} \right|\]
We know that 169 is the perfect square of 13. So, writing the number out of the root we get,
\[\left| d \right| = \left| {\dfrac{{ - 7}}{{13}}} \right|\]
The modulus of the value above is,
\[\left| d \right| = \dfrac{7}{{13}}\]
This is the perpendicular distance of the line from the origin is given by, \[\left| d \right| = \dfrac{7}{{13}}\]
So, the correct answer is “Option A”.

Note: Note that the formula used is simply the way to solve this. Also sometimes in the problem we are asked to find the distance between two straight lines or the distance between a line and a point situated in the same plane. There are different formulas for finding different distances.