Let the system of equations : have infinitely many solutions. Then the radius of the circle centred at and touching the line 4x = 3y is [2025]

Question

Let the system of equations :

$2 x + 3 y + 5 z = 9$

$7 x + 3 y - 2 z = 8$

$12 x + 3 y - (4 + λ) z = 16 - μ$

have infinitely many solutions. Then the radius of the circle centred at $(λ, μ)$ and touching the line 4x = 3y is [2025]

Smart Achievers · Accepted Answer

(2)

Since, the given system of equation has infinitely many solutions.

$△ = △_{1} = △_{2} = △_{3} = 0$

$△ =$ $| \begin{matrix} 2 & 3 & 5 \\ 7 & 3 & - 2 \\ 12 & 3 & - (λ + 4) \end{matrix} |$ $= 0$

$\Rightarrow 12 (- 21) - 3 (- 39) - (λ + 4) (- 15) = 0$

$\Rightarrow - 252 + 117 + 15 (λ + 4) = 0$

$\Rightarrow 15 λ - 75 = 0 \Rightarrow λ = 5$

$△_{1} =$ $| \begin{matrix} 9 & 3 & 5 \\ 8 & 3 & - 2 \\ 16 - μ & 3 & - 9 \end{matrix} |$ $= 0$

$\Rightarrow 9 (- 21) - 3 (- 40 - 2 μ) + 5 (- 24 + 3 μ) = 0$

$\Rightarrow μ = 9$

So, centre of circle (5, 9)

Also, radius = length of $⊥$ from centre (5, 9) to 4x = 3y

$∴ r =$ $| \frac{4 (5) - 3 (9)}{\sqrt{{(4)}^{2} + {(3)}^{2}}} |$ $=$ $| \frac{20 - 27}{5} |$ $= \frac{7}{5}$ .

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