The coordinates of the orthocenter of the triangle formed by the lines $2x^2 – 3xy + y^2$ = 0 and x + y = 1 are :

Question

Accepted Answer

To determine the coordinates of the orthocenter of the triangle formed by the lines given in the problem, we first need to understand the nature of these lines and how they form a triangle.

### Step 1: Identify the Lines

The equation $2x^2 - 3xy + y^2 = 0$ represents a pair of straight lines. To find these lines, we can factor the quadratic expression. This equation can be rewritten as:

$$
(2x - y)(x - y) = 0
$$

This gives us two lines:
1. $2x - y = 0$ or $y = 2x$
2. $x - y = 0$ or $y = x$

The third line given is $x + y = 1$.

### Step 2: Find the Intersection Points

The intersection points of these lines will be the vertices of the triangle.

1. **Intersection of $y = 2x$ and $x + y = 1$:**
   $$
   y = 2x \quad \text{and} \quad x + y = 1
   $$
   Substitute $y = 2x$ into $x + y = 1$:
   $$
   x + 2x = 1 \implies 3x = 1 \implies x = \frac{1}{3}
   $$
   $$
   y = 2 \times \frac{1}{3} = \frac{2}{3}
   $$
   So, the intersection point is $\left(\frac{1}{3}, \frac{2}{3}\right)$.

2. **Intersection of $y = x$ and $x + y = 1$:**
   $$
   y = x \quad \text{and} \quad x + y = 1
   $$
   Substitute $y = x$ into $x + y = 1$:
   $$
   x + x = 1 \implies 2x = 1 \implies x = \frac{1}{2}
   $$
   $$
   y = \frac{1}{2}
   $$
   So, the intersection point is $\left(\frac{1}{2}, \frac{1}{2}\right)$.

3. **Intersection of $y = 2x$ and $y = x$:**
   $$
   2x = x \implies x = 0
   $$
   $$
   y = 0
   $$
   So, the intersection point is $(0, 0)$.

### Step 3: Determine the Orthocenter

The orthocenter of a triangle is the point where the altitudes intersect. For the triangle with vertices $(\frac{1}{3}, \frac{2}{3})$, $(\frac{1}{2}, \frac{1}{2})$, and $(0, 0)$, we can use the properties of the altitudes and the symmetry of the triangle to find the orthocenter.

Given the symmetry and the nature of the lines, the orthocenter can be calculated or observed to be at $\left(\frac{1}{2}, \frac{1}{2}\right)$.

### Conclusion

Therefore, the coordinates of the orthocenter are $\left(\frac{1}{2}, \frac{1}{2}\right)$, which corresponds to Option 3.

### Explanation of Incorrect Options

1. **Option 1: $\left(\frac{1}{4}, \frac{1}{4}\right)$**
   - This point does not satisfy the conditions derived from the intersection of the altitudes of the triangle.

2. **Option 2: $\left(\frac{1}{3}, \frac{1}{3}\right)$**
   - This point is not the intersection of the altitudes and does not align with the calculated orthocenter.

3. **Option 4: (1, 1)**
   - This point is outside the triangle and does not satisfy the conditions for the orthocenter.

Thus, Option 3 is indeed the correct answer.