JEE Main 2025 Jan 28 Shift 2, Mathematics Q17: Introduction to Straight Lines

JEE Main2025Jan 28, Shift 2Mathematics

Two equal sides of an isosceles triangle are along $- x + 2 y = 4$ and $x + y = 4$ . If $m$ is the slope of its third side, then the sum, of all possible distinct values of $m$ , is :

A
$- 6$
B
$12$
C
$6$
D
$- 210$

Solution

The two equal sides have slopes $m_{1} = 1/2$ and $m_{2} = - 1$ . In an isosceles triangle, the third side (base) makes equal angles with the two equal sides, so

$\frac{m - m _{1}}{1 + m m _{1}} = \pm \frac{m - m _{2}}{1 + m m _{2}}$ , i.e. $\frac{m - 1/2}{1 + m /2} = \pm \frac{m + 1}{1 - m}$ .

Cross-multiplying and simplifying both sign cases yields the single quadratic in $m$ that gives non-degenerate solutions: $m^{2} - 6 m - 1 = 0$ .

By Vieta, the sum of roots is $6$ .

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JEE Main 2025 Jan 28 Shift 2, Mathematics Q17: Introduction to Straight Lines

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