Question

The circumcentre of a triangle formed by the lines \[xy + 2x + 2y + 4 = 0\] and \[x + y + 2 = 0\] is
A) \[( - 1, - 1)\]
B) \[(0, - 1)\]
C) \[(1,1)\]
D) \[( - 1,0)\]

Answer 1

Hint: The circumcenter is the center point of the circumcircle drawn around a polygon. The circumcircle of a polygon is the circle that passes through all of its vertices and the center of that circle is called the circumcenter. All polygons that have a circumcircle are known as cyclic polygons. Only regular polygons, triangles, rectangles, and right-kites can have the circumcircle and thus the circumcenter.

Complete step-by-step answer:
Steps to construct the circumcenter of a triangle:
Step 1: Draw the perpendicular bisectors of all the sides of the triangle using a compass.
Step 2: Extend all the perpendicular bisectors to meet at a point. Mark the intersection point as O, this is the circumcenter.
Step 3: Using a compass and keeping O as the center and any vertex of the triangle as a point on the circumference, draw a circle, this circle is our circumcircle whose center is O.

We are given the equations \[xy + 2x + 2y + 4 = 0\] and \[x + y + 2 = 0\]
Consider the equation \[xy + 2x + 2y + 4 = 0\]
This can be rewritten as \[(x + 2)(y + 2) = 0\]
For x = -2 , y = 0
For y = -2 , x = 0
This gives us equations \[x + 2 = 0\] and \[y + 2 = 0\]
Hence, the coordinates of triangle so formed are \[(0, - 2),( - 2,0),( - 2, - 2)\]
Now, these three line equations make a right angle triangle.
Hence, the circumcentre of a triangle will be the midpoint of the hypotenuse.
Midpoint of hypotenuse \[ = \left( {\dfrac{{ - 2 + 0}}{2},\dfrac{{0 - 2}}{2}} \right)\]
 \[ = ( - 1, - 1)\]
Therefore option (1) is the correct answer.
So, the correct answer is “Option 1”.

Note: The circumcenter is the center point of the circumcircle drawn around a polygon. All the vertices of the triangle are equidistant from the circumcenter . when the three line equations make a right angle triangle , the circumcentre of a triangle will be the midpoint of the hypotenuse.