Question

If the lines \( 3x - 4y + 4 = 0 \) and \( 6x - 8y - 7 = 0 \) are the tangents to the same circle, then the area of that circle (in sq. units) is

• A. \( \dfrac{3\pi}{4} \)
• B. \( \dfrac{16\pi}{25} \)
• C. \( \dfrac{9\pi}{4} \)
• D. \( \dfrac{9\pi}{16} \)

Hint:

When two parallel lines are tangents to a circle, the perpendicular distance between them is twice the radius of the circle.

Answer 1

The Correct Option is D

Step 1: Find the distance between the given parallel lines.
The two lines \( 3x - 4y + 4 = 0 \) and \( 6x - 8y - 7 = 0 \) can be rewritten as: \[ \text{Line 1: } \frac{3x - 4y + 4}{5} = 0
\text{and}
\text{Line 2: } \frac{6x - 8y - 7}{10} = 0 \] Dividing the second line by 2 gives: \[ 3x - 4y - \frac{7}{2} = 0 \] Now, compute the distance \( d \) between the two parallel lines: \[ d = \frac{\left| \left( -\frac{7}{2} \right) - 4 \right|}{\sqrt{3^2 + (-4)^2}} = \frac{\left| -\frac{15}{2} \right|}{5} = \frac{15}{10} = \frac{3}{2} \] Step 2: Use the radius to compute area.
If these are tangents to a circle on opposite sides, then the distance between them is \( 2r \), so: \[ 2r = \frac{3}{2} \Rightarrow r = \frac{3}{4} \] Area \( A \) of the circle is: \[ A = \pi r^2 = \pi \left(\frac{3}{4}\right)^2 = \frac{9\pi}{16} \]