Answer
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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.
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.
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