JEE Main · 2026 · 24 Jan 2026 · Shift 1 · Straight Lines

Let

A (1, 0), B (2, - 1)

and

C (\frac{7}{3}, \frac{4}{3})

be three points. If the equation of the bisector of the angle ABC is

α x + β y = 5

, then the value of

α^{2} + β^{2}

(A)8
(B)5
(C)13
(D)10✓ Correct

Answer: (D)

Solution

The internal bisector of

∠ A B C

runs along the sum of the unit vectors from

B

toward

A

and toward

C

. Here

B A = (- 1, 1)

with

∣ B A ∣ = 2

, and

B C = (\frac{1}{3}, \frac{7}{3})

with

∣ B C ∣ = \frac{5 2}{3}

. Their unit sum is proportional to

(- 1, 3)

, so the bisector through

B (2, - 1)

has slope

- 3

y + 1 = - 3 (x - 2)

, i.e.

3 x + y = 5

. Matching

α x + β y = 5

gives

α = 3, β = 1

, so

α^{2} + β^{2} = 10

JEE Main · 2026 · 24 Jan 2026 · Shift 1 · Single correct option

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