Question

The three lines of a triangle are given by \((x^2 - y^2)(2x + 3y - 6) = 0\). If the point \((-2,\lambda)\) lies inside and \((\mu,1)\) lies outside the triangle, then

• A. \(\lambda \in (1,\frac{10}{3}),\ \mu \in (-3,5)\)
• B. \(\lambda \in (2,\frac{10}{3}),\ \mu \in (-1,1)\)
• C. \(\lambda \in (-1,\frac{9}{2}),\ \mu \in (-2,\frac{10}{3})\)
• D. None of the above

Hint:

For triangle region problems, use sign method for all three lines.

Answer 1

The Correct Option is D

To solve this problem, we need to analyze the given equation representing the lines of a triangle and find out the conditions under which the point \((-2, \lambda)\) lies inside the triangle, and the point \((\mu, 1)\) lies outside the triangle.

The equation given is \((x^2 - y^2)(2x + 3y - 6) = 0\). This implies three lines:

• \(x^2 - y^2 = 0\), which can be factored as \((x-y)(x+y) = 0\).
• \(2x + 3y - 6 = 0\).

Thus, the lines are:

• \(x - y = 0\), or \(y = x\).
• \(x + y = 0\), or \(y = -x\).
• \(2x + 3y = 6\).

Our task is to determine the regions formed by these lines. The lines \(y = x\) and \(y = -x\) intersect at the origin (0,0). The line \(2x + 3y = 6\) is a straight line that can be rearranged to \(y = -\frac{2}{3}x + 2\).

To determine whether a point lies inside or outside the triangle formed by these three lines, we evaluate the sign of each expression with respect to the point.

For point \((-2, \lambda)\) to lie inside:

• It should satisfy \((x - y)(x + y)(2x + 3y - 6) < 0\).

For point \((\mu, 1)\) to lie outside:

• This point should not satisfy \((x - y)(x + y)(2x + 3y - 6) < 0\).

Combining these, we solve strategically:

1. Consider \((-2, \lambda)\) in the expressions for all lines. It must yield:
   • \(-2 - \lambda \neq 0\)
   • \(-2 + \lambda \neq 0\)
   • \(-4 + 3\lambda - 6 < 0 \Rightarrow 3\lambda < 10 \Rightarrow \lambda < \frac{10}{3}\)
2. For point \((\mu, 1)\):
   • \(\mu - 1\neq 0\)
   • \(\mu + 1\neq 0\)
   • \(2\mu + 3 \cdot 1 - 6 \geq 0 \Rightarrow 2\mu \geq 3\Rightarrow \mu \geq \frac{3}{2}\)

Comparing these conditions with the given options:

• Option (1): \(\lambda \in (1,\frac{10}{3}), \mu \in (-3,5)\)
• Option (2): \(\lambda \in (2,\frac{10}{3}), \mu \in (-1,1)\)
• Option (3): \(\lambda \in (-1,\frac{9}{2}), \mu \in (-2,\frac{10}{3})\)
• Option (4): None of the Above

Since none of these range combinations precisely matches both derived conditions: \(\lambda < \frac{10}{3}\) and \(\mu \geq \frac{3}{2}\), thus, the correct answer is:

None of the Above