Question

The circumcenter 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

The correct option is A

$(-1,-1)$

Explanation for the correct option:

Step 1: Finding the co-ordinates

Given, the equation of lines by which the triangle is formed are: $\mathrm{xy}+2\mathrm{x}+2\mathrm{y}+4=0$ and $x+y+2=0$

Re-writing the given equation of lines, we get

$$\begin{gathered} \Rightarrow xy+2x+2y+4=0 \\ \Rightarrow (x+2)(y+2)=0 \end{gathered}$$

Therefore, $x+2=0$ or $y+2=0$.

Hence, $\mathrm{x}=-2$, $y=-2$ and $\mathrm{x}+\mathrm{y}+2=0$ are equations of the line.

On solving the equation $\mathrm{x}+\mathrm{y}+2=0$ and $\mathrm{x}=-2$, we get $y=0$.

Therefore, the point of intersection is $\left(-2,0\right)$.

On solving the equation $\mathrm{x}+\mathrm{y}+2=0$ and $y=-2$, we get $x=0$.

Therefore, the point of intersection is $\left(0,-2\right)$.

On solving the equation $y=-2$ and $\mathrm{x}=-2$, the point of intersection is $\left(-2,0\right)$.

Hence, the coordinates of triangle are $(0,-2),(-2,0),(-2,-2)$.

Step 2: Finding the circumcenter

Now, these three lines make a right-angled triangle.

So, the circumcenter of the right-angled triangle is at the midpoint of the hypotenuse.

Hence, the midpoint and circumcenter lie on a single point.

The midpoint formula is given by $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.

Here the endpoints of hypotenuse are $\left(-2,0\right),\left(0,-2\right)$.

$$\begin{gathered} \therefore M\mathrm{idpoint}=\left(\frac{(-2+0)}{2},\frac{(0-2)}{2}\right) \\ =\left(-1,-1\right) \end{gathered}$$

The circumcentre of the triangle is $\left(-1,-1\right)$.

Therefore, option (A) is the correct answer.