Question

The area (in square unit) of the circle, which touches the lines \[4x + 3y = 15\] and \[4x + 3y = 5\] is \[m\pi \]. Find \[m\].

Answer 1

Hint: First we will have the diameter, which is same as the perpendicular distance between two lines, \[ax + by + c = 0\] and \[ax + by + d = 0\] is\[\dfrac{{\left| {c - d} \right|}}{{\sqrt {{a^2} + {b^2}} }}\] units. Then we will draw the diagram and then find the value of \[a\], \[b\], \[c\], and \[d\] in the formula. Then we will use the given conditions to find the required value.

Complete step-by-step answer:
We are given that the area (in square unit) of the circle, which touches the lines \[4x + 3y = 15\] and \[4x + 3y = 5\] is \[m\pi \].

Rewriting the given equation, we get
\[4x + 3y - 15 = 0{\text{ ......eq.(1)}}\]
\[4x + 3y - 5 = 0{\text{ ......eq.(2)}}\]
Since it is clear that the given lines are parallel, so we will have the diameter, which is same as the perpendicular distance between two lines, \[ax + by + c = 0\] and \[ax + by + d = 0\] is\[\dfrac{{\left| {c - d} \right|}}{{\sqrt {{a^2} + {b^2}} }}\] units.
Finding the value of \[a\], \[b\], \[c\], and \[d\] from the equations (1) and (2), we get
\[a = 4\]
\[b = 3\]
\[c = - 15\]
\[d = - 5\]
Substituting the value of \[a\], \[b\], \[c\], and \[d\] in the formula of perpendicular distance between two lines, we get
\[
   \Rightarrow \dfrac{{\left| { - 15 - \left( { - 5} \right)} \right|}}{{\sqrt {{4^2} + {3^2}} }} \\
   \Rightarrow \dfrac{{\left| { - 15 + 5} \right|}}{{\sqrt {16 + 9} }} \\
   \Rightarrow \dfrac{{\left| { - 10} \right|}}{{\sqrt {25} }} \\
   \Rightarrow \dfrac{{10}}{5} \\
   \Rightarrow 2{\text{ units}} \\
 \]
So, the diameter is 2 units.
Dividing the above diameter by 2 to find the radius of the circle, we get
\[ \Rightarrow \dfrac{2}{2} = 1{\text{ units}}\]
Using the formula of area of circle is ,\[\pi {r^2}\] where \[r\] is the radius, we get
\[
   \Rightarrow \pi {\left( 1 \right)^2} \\
   \Rightarrow \pi \left( 1 \right) \\
   \Rightarrow \pi {\text{ units}} \\
 \]
So, we have according to the problem is \[m\pi = \pi \].
Dividing the above equation by \[\pi \] on both sides, we get
\[
   \Rightarrow \dfrac{{m\pi }}{\pi } = \dfrac{\pi }{\pi } \\
   \Rightarrow m = 1 \\
 \]

Therefore, the required value is 1.

Note: We know that the perpendicular distance formula of the lines is used and we see that the perpendicular distance between two lines, \[ax + by + c = 0\] and \[ax + by + d = 0\] is\[\dfrac{{\left| {c - d} \right|}}{{\sqrt {{a^2} + {b^2}} }}\]. Also, we are supposed to avoid calculations. We have to find the radius, do not solve using the diameter or else the answer will be wrong. Diagrams will help in better understanding.