Question:
The circumcenter of a triangle formed by the lines \( xy + 2x + 2y + 4 = 0 \) and \( x + y + 2 = 0 \) is
The circumcenter of a triangle formed by the lines \( xy + 2x + 2y + 4 = 0 \) and \( x + y + 2 = 0 \) is
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To find the circumcenter of a right triangle, find the midpoint of the hypotenuse, as it is the point where the perpendicular bisectors of the sides meet.
Updated On: Apr 22, 2026
- \( (0, -1) \)
- \( (-1, 0) \)
- \( (1, 1) \)
- \( (-1, -1) \)
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The Correct Option is D
Solution and Explanation
Step 1: Understand the problem.
We are given two lines that form a triangle. We need to find the circumcenter of the triangle formed by these lines. The circumcenter is the point where the perpendicular bisectors of the sides of the triangle intersect.
Step 2: Find the equations of the lines.
The two lines given are: 1. \( xy + 2x + 2y + 4 = 0 \)
2. \( x + y + 2 = 0 \)
We need to find the intersection of these lines to locate the vertices of the triangle.
Step 3: Solve the system of equations.
Solve the system of equations to find the intersection points of the lines: From the second equation, \( x + y = -2 \), we get: \[ y = -x - 2 \] Substitute this into the first equation \( xy + 2x + 2y + 4 = 0 \): \[ x(-x - 2) + 2x + 2(-x - 2) + 4 = 0 \] Simplify the equation: \[ -x^2 - 2x + 2x - 2x - 4 + 4 = 0 \] \[ -x^2 - 2x = 0 \] \[ x(x + 2) = 0 \] Thus, \( x = 0 \) or \( x = -2 \). Substituting these into \( y = -x - 2 \), we get:
- For \( x = 0 \), \( y = -2 \)
- For \( x = -2 \), \( y = 0 \) So, the two intersection points are \( (0, -2) \) and \( (-2, 0) \).
Step 4: Find the circumcenter.
The circumcenter of a right triangle is the midpoint of the hypotenuse. The hypotenuse is the line segment connecting the points \( (0, -2) \) and \( (-2, 0) \). The midpoint is: \[ \left( \frac{0 + (-2)}{2}, \frac{-2 + 0}{2} \right) = (-1, -1) \]
Step 5: Conclusion.
Thus, the circumcenter of the triangle is \( (-1, -1) \), corresponding to option (D).
Step 6: Verification.
We verify the calculation by checking the properties of the circumcenter. The point \( (-1, -1) \) lies on the perpendicular bisector of the hypotenuse, confirming that it is the circumcenter.
We are given two lines that form a triangle. We need to find the circumcenter of the triangle formed by these lines. The circumcenter is the point where the perpendicular bisectors of the sides of the triangle intersect.
Step 2: Find the equations of the lines.
The two lines given are: 1. \( xy + 2x + 2y + 4 = 0 \)
2. \( x + y + 2 = 0 \)
We need to find the intersection of these lines to locate the vertices of the triangle.
Step 3: Solve the system of equations.
Solve the system of equations to find the intersection points of the lines: From the second equation, \( x + y = -2 \), we get: \[ y = -x - 2 \] Substitute this into the first equation \( xy + 2x + 2y + 4 = 0 \): \[ x(-x - 2) + 2x + 2(-x - 2) + 4 = 0 \] Simplify the equation: \[ -x^2 - 2x + 2x - 2x - 4 + 4 = 0 \] \[ -x^2 - 2x = 0 \] \[ x(x + 2) = 0 \] Thus, \( x = 0 \) or \( x = -2 \). Substituting these into \( y = -x - 2 \), we get:
- For \( x = 0 \), \( y = -2 \)
- For \( x = -2 \), \( y = 0 \) So, the two intersection points are \( (0, -2) \) and \( (-2, 0) \).
Step 4: Find the circumcenter.
The circumcenter of a right triangle is the midpoint of the hypotenuse. The hypotenuse is the line segment connecting the points \( (0, -2) \) and \( (-2, 0) \). The midpoint is: \[ \left( \frac{0 + (-2)}{2}, \frac{-2 + 0}{2} \right) = (-1, -1) \]
Step 5: Conclusion.
Thus, the circumcenter of the triangle is \( (-1, -1) \), corresponding to option (D).
Step 6: Verification.
We verify the calculation by checking the properties of the circumcenter. The point \( (-1, -1) \) lies on the perpendicular bisector of the hypotenuse, confirming that it is the circumcenter.
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