Question

Consider a triangle \(Δ\) whose two sides lie on the x-axis and the line \(x + y + 1 = 0\). If the orthocenter of \(Δ\) is (1, 1), then the equation of the circle passing through the vertices of the triangle \(Δ\) is;

• A. x² + y² – 3x + y = 0
• B. x² + y² + x + 3y = 0
• C. x² + y² + 2y – 1 = 0
• D. x² + y² + x + y = 0

Answer 1

The Correct Option is B

Step 1: Understanding the problem
We are given a triangle \( \Delta \) whose two sides lie on the x-axis and the line \( x + y + 1 = 0 \).
The orthocenter of the triangle is given as \( (1, 1) \). We are tasked with finding the equation of the circle passing through the vertices of the triangle \( \Delta \).
Step 2: Key properties and setup
In a triangle, the orthocenter is the point where the altitudes meet. The circumcenter lies on the perpendicular bisector of the sides of the triangle, and the orthocenter, circumcenter, and the reflection of the orthocenter across the circumcenter will lie on the circumcircle.
To solve for the equation of the circumcircle, we need to consider the mirror image of the orthocenter with respect to the line \( x + y + 1 = 0 \) and the x-axis.
Step 3: Reflection of the orthocenter
The image of the orthocenter \( (1, 1) \) reflected over the x-axis is \( (1, -1) \).
The image of \( (1, 1) \) reflected over the line \( x + y + 1 = 0 \) is \( (-2, -2) \).
Step 4: Circle passing through the points
The circle passing through the points \( (1, -1) \) and \( (-2, -2) \) must be the circumcircle of the triangle.
To determine this circle, we calculate the equation of the circle passing through both points \( (1, -1) \) and \( (-2, -2) \).
Step 5: Equation of the circle
After reflecting the points and finding the required circumcenter, the equation of the circle passing through the vertices of the triangle is:
\[ x^2 + y^2 + x + 3y = 0 \] Therefore, the correct equation of the circle is \( x^2 + y^2 + x + 3y = 0 \).
Step 6: Conclusion
Thus, the correct option is: B.