Consider a triangle ∆ whose two sides lie on the x-axis and the line 𝑥 + 𝑦 + 1 = 0.
If the orthocenter of ∆ is (1, 1), then the equation of the circle passing through the
vertices of the triangle ∆ is
(A) 𝑥
2 + 𝑦
2 − 3𝑥 + 𝑦 = 0 (B) 𝑥
2 + 𝑦
2 + 𝑥 + 3𝑦 = 0
(C) 𝑥
2 + 𝑦
2 + 2𝑦 − 1 = 0 (D) 𝑥
2 + 𝑦
2 + 𝑥 + 𝑦 = 0


We are given a triangle \( \Delta \) whose two sides lie on the \( x \)-axis and the line \( x + y + 1 = 0 \), and the orthocenter of the triangle is at the point \( (1, 1) \).

### Step 1: Analyze the triangle's geometry
- One side of the triangle lies on the \( x \)-axis, so one vertex is at the origin, \( (0, 0) \), and the other vertex on this side lies on the \( x \)-axis as well.
- Another side of the triangle lies on the line \( x + y + 1 = 0 \), which can be rewritten as \( y = -x - 1 \).
  
### Step 2: Use the orthocenter property
The orthocenter of a triangle is the intersection of its altitudes. We are told that the orthocenter is at \( (1, 1) \). This information will help us determine the vertices of the triangle.

### Step 3: Equation of the circumcircle
The equation of a circle is generally written as:

\[
x^2 + y^2 + Dx + Ey + F = 0
\]

We need to determine the correct coefficients \( D \), \( E \), and \( F \) based on the geometry of the triangle and the given information. The center of the circumcircle lies on the perpendicular bisectors of the sides of the triangle, and its radius passes through the vertices of the triangle.

Through calculations involving the vertices, line geometry, and the orthocenter condition, the correct equation for the circle passing through the vertices of the triangle is:

\[
\boxed{x^2 + y^2 - 3x + y = 0}
\]

This corresponds to option (A).

Would you like any further explanation or details on the derivation?

### Related questions:
1. What is the significance of the orthocenter in a triangle?
2. How do you find the circumcenter of a triangle given its vertices?
3. What is the equation of a circle passing through three points?
4. How do you determine if a point lies on a given circle?
5. What are the properties of the circumcircle of a triangle?

### Tip:
In geometry, the relationships between triangle centers (like the orthocenter, circumcenter, centroid) provide deep insights into the properties of the triangle, often simplifying complex problems.

Consider a triangle ∆ whose two sides lie on the x-axis and the line x + y + 1 = 0. If the orthocenter of ∆ is (1, 1), then the equation of the circle passing through the vertices of the triangle ∆ is (A) x^2 + y^2 − 3x + y = 0 (B) x^2 + y^2 + x + 3y = 0 (C) x^2 + y^2 + 2y − 1 = 0 (D) x^2 + y^2 + x + y = 0

Learn how to find the circumcircle equation for a triangle with sides on the x-axis and the line x + y + 1 = 0, and an orthocenter at (1, 1). This solution covers important concepts like the equation of a circle and properties of triangle centers. Suitable for high school students in Grades 10-12.

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