Question

The area (in square unit) of the circle which touches the lines \(4x+3y=15\) and \(4x+3y=5\) is

• A. \(4\pi\)
• B. \(2\pi\)
• C. \(\pi\)
• D. \(\pi\)

Hint:

For a circle touching two parallel lines, the distance between lines equals diameter of circle.

Answer 1

The Correct Option is D

Step 1: Identify the two parallel lines.
\[ 4x+3y=15,\quad 4x+3y=5 \]
They are parallel since coefficients of \(x,y\) are same.
Step 2: Distance between two parallel lines.
Write both in standard form:
\[ 4x+3y-15=0 \]
\[ 4x+3y-5=0 \]
Distance:
\[ d=\frac{|(-15)-(-5)|}{\sqrt{4^2+3^2}} =\frac{| -10 |}{5}=2 \]
Step 3: Radius of the circle touching both lines.
If circle touches both parallel lines, its diameter equals distance.
\[ 2r=d \Rightarrow r=1 \]
Step 4: Area of circle.
\[ A=\pi r^2=\pi(1)^2=\pi \]
Final Answer:
\[ \boxed{\pi} \]