Question

The lengths of two equal sides of an isosceles triangle are given by \(L_1 = 2x + y - 3 = 0\) and \(L_2 = ax + by + c = 0\). If \(L_3 = x + 2y + 1 = 0\) is the third side of this triangle and \((5, 1)\) is a point on \(L_2\), then \(b^2/|ac|\) is:

• A. \(\frac{121}{2}\)
• B. \(\frac{49}{52}\)
• C. \(\frac{81}{49}\)
• D. \(\frac{25}{4}\)

Hint:

Use the properties of isosceles triangles and the equation of lines to solve for unknown parameters.
- For triangles, consider the geometry of points and lines to derive the necessary relations.

Answer 1

The Correct Option is A

Step 1: Given conditions.
From the equation \(L_1 = 2x + y - 3 = 0\), we know the coordinates of point \(A\) lie on this line. The second equation \(L_2 = ax + by + c = 0\) represents the other equal side, and the third equation \(L_3 = x + 2y + 1 = 0\) represents the third side. The coordinates of the point \((5, 1)\) lie on \(L_2\), so substituting this point into the equation of \(L_2\), we get: \[ a(5) + b(1) + c = 0 \quad \Rightarrow \quad 5a + b + c = 0 \quad \cdots(1). \] Step 2: Solving for the value of \(b^2/|ac|\).
By applying the equations and properties of the given triangle, we solve for \(b^2/|ac|\) and obtain the final answer: \[ \boxed{\frac{121}{2}}. \]