`x+y=7` and `a x^2+2h x y+a y^2=0,(a!=0)` , are three real distinct lines forming a triangle. Then the triangle is isosceles (b) scalene equilateral (d) right angled

Options

A. isosceles
B. scalene
C. equilateral
D. right - angled

Explanation

To solve the problem, we need to analyze the given equations and determine the nature of the triangle formed by the lines represented by these equations.

### Step-by-Step Solution:

1. **Identify the Given Lines:**
We have two equations:
- Line 1: \( x + y = 7 \)
- Line 2: \( ax^2 + 2hxy + ay^2 = 0 \) (where \( a \neq 0 \))

2. **Understand the Second Equation:**
The second equation represents a pair of straight lines. We can rewrite it in a more familiar form. This quadratic equation can be factored to find the slopes of the lines it represents.

3. **Finding the Slopes of the Lines:**
From the equation \( ax^2 + 2hxy + ay^2 = 0 \), we can use the formula for the slopes of the lines:
\[ m_1, m_2 = \frac{-h \pm \sqrt{h^2 - a^2}}{a} \]
This gives us the slopes of the two lines represented by the quadratic equation.

4. **Finding the Intersection Points:**
To find the points of intersection of the lines, we substitute \( y = 7 - x \) (from Line 1) into the second equation:
\[ ax^2 + 2hx(7 - x) + a(7 - x)^2 = 0 \]
Simplifying this will give us a quadratic equation in terms of \( x \).

5. **Analyzing the Triangle Formed:**
The intersection points of the lines will give us the vertices of the triangle. We need to check the lengths of the sides of the triangle formed by these intersection points.

6. **Determine the Nature of the Triangle:**
- If two sides are equal, the triangle is isosceles.
- If all sides are different, the triangle is scalene.
- If one angle is \( 90^\circ \), the triangle is right-angled.
- If all angles are \( 60^\circ \), the triangle is equilateral.

In this case, since we have determined that the two lines from the quadratic equation are symmetric about the line \( x = y \) (which is the angle bisector), we can conclude that the triangle formed by these lines and the line \( x + y = 7 \) is isosceles.

### Conclusion:
The triangle formed by the lines \( x + y = 7 \) and the lines represented by \( ax^2 + 2hxy + ay^2 = 0 \) is an **isosceles triangle**.