Multiple choice

If $L_{1}$ and $L_{2}$ are the length of the tangent from $(0, 5)$ to the circles $x^{2}+y^{2}+2x-4=0$ and $x^{2}+y^{2}-y+1=0$ then

  1. $L_{1}=2L_{2}$
  2. $L_{2}=2L_{1}$
  3. $L_{1}=L_{2}$
  4. $L_{1}^{2}=L_{2}$
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C Correct answer
AI explanation

Using the tangent length formula L = sqrt(x1^2 + y1^2 + 2gx1 + 2fy1 + c), we substitute the point (0, 5) into the first circle's equation x^2 + y^2 + 2x - 4 = 0 to get L1 = sqrt(0^2 + 5^2 + 0 - 4) = sqrt(21). For the second circle x^2 + y^2 - y + 1 = 0, the length is L2 = sqrt(0^2 + 5^2 - 5 + 1) = sqrt(21). Therefore, L1 equals L2.

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