Multiple choice

lf the circles $(\mathrm{x}-\mathrm{a})^{2}+(\mathrm{y}-\mathrm{b})^{2}=\mathrm{r}^{2},\ (\mathrm{x}-\mathrm{b})^{2}+(\mathrm{y}- \mathrm{a})^{2}=\mathrm{r}^{2}$ have three common tangents, then which of the following relation follows:

  1. $(a-b)^{2}$ $=2\mathrm{r}^{2}$
  2. $(\mathrm{a}+\mathrm{b})^{2}=2\mathrm{r}^{2}$
  3. $\mathrm{a}^{2}+\mathrm{b}^{2}=2\mathrm{r}^{2}$
  4. $\mathrm{a}^{2}+\mathrm{b}^{2}=\mathrm{r}^{2}$
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A Correct answer
Explanation

For two circles with equal radii r to have three common tangents, they must be externally tangent. The distance between centers (a, b) and (b, a) must equal 2r. Distance = sqrt((a-b)^2 + (b-a)^2) = sqrt(2(a-b)^2) = |a-b|*sqrt(2). Setting this to 2r and squaring gives 2(a-b)^2 = 4r^2, or (a-b)^2 = 2r^2.

AI explanation

The centers of the two circles are (a, b) and (b, a), and both have radius r. The distance between the centers is the square root of the quantity (a minus b) squared plus (b minus a) squared, which simplifies to the square root of 2 times (a minus b) squared. Since the circles have exactly three common tangents, they touch externally, meaning the distance between centers equals r plus r, or 2r. Squaring 2r gives 4r squared, and equating this to 2 times (a minus b) squared yields the relation (a minus b) squared equals 2r squared.

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