Straight Lines
Straight Lines
nta_pyq_2025_jan
Grade 11
Question:
Two equal sides of an isosceles triangle are along -x + 2y = 4 and x + y = 4. If m is the slope of its third side, then the sum, of all possible distinct values of m, is :
Step-by-Step Solution
Key Concept: Apply the core result for lines, slopes and distances and simplify using the given constraints.
(3) Slope of the third side = slope of the perpendicular bisector of given lines -x+2y-4 x+y-4 h : = $\pm$ \sqrt5 \sqrt2 h1 : \sqrt2(-x + 2y - 4) = \sqrt5(x + y - 4) h2 : \sqrt2(-x + 2y - 4) = -\sqrt5(x + y - 4) \sqrt5 + \sqrt2 ML : -[ ] 1 \sqrt 5 - 2\sqrt 2 \sqrt5 - \sqrt2 ML : -[ ] 2 \sqrt 5 + 2\sqrt 2 \sqrt5 + \sqrt2 \sqrt5 - \sqrt2 ML + ML = -[ + ] 1 2 \sqrt5 - 2\sqrt2 \sqrt5 + 2\sqrt2 (\sqrt5 + \sqrt2)(\sqrt5 + 2\sqrt2) + (\sqrt5 - \sqrt2)(\sqrt5 - 2\sqrt2) = -[ ] -3 = 6
Correct Answer: 3