JEE Main · 2025 · 28 Jan 2025 · Shift 2 · Straight Lines

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✓ Correct
(D) $- 210$

Answer: (C)

Solution

The two equal sides lie along

- x + 2 y = 4

(slope

\frac{1}{2}

) and

x + y = 4

(slope

- 1

). For the base with slope

m

, the isosceles condition requires equal base angles, i.e.

\frac{m - \frac{1}{2}}{1 + \frac{m}{2}} = \frac{m + 1}{1 - m}

i.e.

\frac{2 m - 1}{2 + m} = \frac{m + 1}{1 - m}

Taking the positive sign gives

m^{2} = - 1

(no real solution). Taking the negative sign:

(2 m - 1) (1 - m) = - (m + 1) (2 + m)

- 2 m^{2} + 3 m - 1 = - m^{2} - 3 m - 2

m^{2} - 6 m - 1 = 0

By Vieta's formulas, the sum of the two possible slopes is 6.

JEE Main · 2025 · 28 Jan 2025 · Shift 2 · Single correct option

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