JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let $(\alpha, \beta)$ be the centroid of the triangle formed by the lines $15x - y = 82$ , $6x - 5y = -4$ and $9x + 4y = 17$ . Then, $\alpha + 2\beta$ and $2\alpha - \beta$ are the roots of the equation:

Choose the correct answer:

Explanation

Solution:

Step 1 (Vertices): Lines ko pair mein solve karke vertices nikaalein:
- $(15x - y = 82) \cap (6x - 5y = -4) \implies A(6, 8)$
- $(6x - 5y = -4) \cap (9x + 4y = 17) \implies B(1, 2)$
- $(15x - y = 82) \cap (9x + 4y = 17) \implies C(5, -7)$
Step 2 (Centroid): $\alpha = \frac{6+1+5}{3} = 4$ aur $\beta = \frac{8+2-7}{3} = 1$ .
Step 3 (Roots):

$r_1 = \alpha + 2\beta = 4 + 2(1) = 6$

$r_2 = 2\alpha - \beta = 2(4) - 1 = 7$
Step 4 (Equation): Roots $6$ aur $7$ wali equation:

$x^2 - (6+7)x + (6 \times 7) = 0 \implies x^2 - 13x + 42 = 0$