Multiple choice

Two circles, each radius $5$, have a common tangent at $(1,1)$ whose equation is $3x+4y-7=0$. Then their centre are

  1. $(4,-5),(-2,3)$
  2. $(4,-3),(-2,5)$
  3. $(4,5),(-2,-3)$
  4. $None\ of\ these$
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C Correct answer
Explanation

The centers lie on a line perpendicular to the tangent 3x + 4y - 7 = 0, which has slope -3/4. The normal line has slope 4/3 and passes through (1,1), so y - 1 = 4/3(x - 1) => 4x - 3y - 1 = 0. Checking the options, (4,5) and (-2,-3) satisfy this line equation and are distance 5 from (1,1).

AI explanation

The slope of the tangent line 3x + 4y - 7 = 0 is -3/4, meaning the line segment connecting the two centers has a slope of 4/3. Since the circles have a radius of 5 and touch at the point (1,1), the distance from (1,1) to each center must be 5. Using the point-slope form for the line of centers, the direction vector with slope 4/3 is utilized, and moving a distance of 5 from the tangent point in both directions places the centers at (4,5) and (-2,-3).

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