Multiple choice

The Circumcentre of the triangle formed by the lines $xy+2x+2y+4=0$ and $x+y+2=0$ is _________ .

  1. $(-1,-1)$
  2. $(-1,-2)$
  3. $(-1,1)$
  4. $(1,2)$
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A Correct answer
Explanation

The first equation xy+2x+2y+4=0 factors as (x+2)(y+2)=0, representing lines x=-2 and y=-2. The second line is x+y+2=0. The triangle vertices are (-2,-2), (-2,0), and (0,-2). The circumcentre of a right triangle is the midpoint of the hypotenuse, which is (-1,-1).

AI explanation

The product of the equations x + y + 2 = 0 and xy + 2x + 2y + 4 = 0 gives xy(x+y+2) + 2(x+y+2)^2 = 0, which simplifies to x^2 + 2xy + y^2 + 5xy + 4x + 4y + 4 = 0. This conic represents two perpendicular lines intersecting at (-1,-1), which form a right-angled triangle with the line x + y + 2 = 0. The circumcentre of a right-angled triangle is its right-angled vertex, which is (-1,-1).

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