The three sides of a triangle are given by `(x^2 - y^2)(2x+3y-6) = 0`. If the points (-2,a) lies inside and (b,1) lies outside the triangle, then

A. `2ltalt(10)/(3)`
B. `-2ltalt(10)/(3)`
C. `-1ltblt(9)/(2)`
D. `-1ltblt1`

Options

A. `2ltalt(10)/(3)`
B. `-2ltalt(10)/(3)`
C. `-1ltblt(9)/(2)`
D. `-1ltblt1`

Answer

To solve the problem, we need to analyze the given equation of the triangle and the conditions for the points (-2, a) and (b, 1).

### Step-by-Step Solution:

1. **Identify the Triangle's Sides**:
The equation given is \((x^2 - y^2)(2x + 3y - 6) = 0\).
This can be factored into:
\[ (x - y)(x + y)(2x + 3y - 6) = 0 \]
This gives us three lines:
- \(x - y = 0\) (or \(y = x\))
- \(x + y = 0\) (or \(y = -x\))
- \(2x + 3y - 6 = 0\) (or \(y = \frac{6 - 2x}{3}\))

2. **Graph the Lines**:
- The line \(y = x\) passes through the origin and has a slope of 1.
- The line \(y = -x\) also passes through the origin but has a slope of -1.
- The line \(y = \frac{6 - 2x}{3}\) intersects the y-axis at \(y = 2\) (when \(x = 0\)) and the x-axis at \(x = 3\) (when \(y = 0\)).

3. **Determine the Triangle's Vertices**:
The intersection points of the lines will give us the vertices of the triangle:
- Intersection of \(y = x\) and \(y = \frac{6 - 2x}{3}\):
\[ x = \frac{6 - 2x}{3} \implies 3x = 6 - 2x \implies 5x = 6 \implies x = \frac{6}{5}, y = \frac{6}{5} \]
- Intersection of \(y = -x\) and \(y = \frac{6 - 2x}{3}\):
\[ -x = \frac{6 - 2x}{3} \implies -3x = 6 - 2x \implies -x = 6 \implies x = -6, y = 6 \]
- Intersection of \(y = x\) and \(y = -x\) gives the origin (0, 0).

4. **Check the Points**:
- For the point \((-2, a)\) to lie inside the triangle:
- Substitute into the line equations:
- From \(y = \frac{6 - 2(-2)}{3}\):
\[ 3a < 10 \implies a < \frac{10}{3} \]
- From \(y = x\):
\[ -2 + a > 0 \implies a > 2 \]
- Thus, \(2 < a < \frac{10}{3}\).

- For the point \((b, 1)\) to lie outside the triangle:
- From \(y = x\):
\[ b + 1 > 0 \implies b > -1 \]
- From \(y = -x\):
\[ b - 1 < 0 \implies b < 1 \]
- Thus, \(-1 < b < 1\).

5. **Final Conditions**:
- For \(a\): \(2 < a < \frac{10}{3}\)
- For \(b\): \(-1 < b < 1\)

### Conclusion:
Based on the conditions derived:
- The correct answer options are:
- A: \(2 < a < \frac{10}{3}\)
- D: \(-1 < b < 1\)