Multiple choice

The lines $3x-4y+\lambda=0$ and $6x-8+\mu=0$ are tangent to the same circle. The radius of the circle is

  1. $\left|\dfrac {2\lambda-\mu}{20}\right|$
  2. $\left|\dfrac {2\mu+\lambda}{20}\right|$
  3. $\left|\dfrac {\lambda-\mu}{20}\right|$
  4. $None\ of\ these$
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A Correct answer
AI explanation

The second line equation must be read as 6x - 8y + mu = 0, which simplifies to 3x - 4y + mu/2 = 0, showing that both lines are parallel. For a circle to be tangent to both parallel lines, its diameter must equal the perpendicular distance between the lines. Calculating the distance between the lines using the point-to-line formula gives |lambda - mu/2| divided by sqrt(3^2 + (-4)^2), which equals |2*lambda - mu|/10, making the radius half of that distance, or |2*lambda - mu|/20.

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