Multiple choice

The circumcentre of the triangle formed by the lines $\mathrm{x}^{2}-\mathrm{y}^{2}=0$ and $\mathrm{y}-5=0$ is

  1. $(5, 0)$
  2. $(0, 5)$
  3. $(0, 0)$
  4. $(5, 5)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The lines are x=y, x=-y, and y=5. The vertices of the triangle are (0,0), (5,5), and (-5,5). This is a right-angled triangle at (0,0). The circumcenter of a right triangle is the midpoint of the hypotenuse. The hypotenuse connects (5,5) and (-5,5), so the midpoint is (0,5).

AI explanation

The equation $x^2 - y^2 = 0$ factors into the lines $y = x$ and $y = -x$, which intersect at the origin $(0,0)$. The third line $y - 5 = 0$ is a horizontal line that intersects $y = x$ at $(5,5)$ and $y = -x$ at $(-5,5)$. The circumcentre of this isosceles right triangle lies on the y-axis at the midpoint of the hypotenuse connecting $(5,5)$ and $(-5,5)$, which is $(0,5)$.

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