Let the equation x(x + 2)(12 – k) = 2 have equal roots. Then the distance of the point from the line 3x + 4y + 5 = 0 is [2025]

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Solution

Q.
Let the equation x(x + 2)(12 – k) = 2 have equal roots. Then the distance of the point $(k, \frac{k}{2})$ from the line 3x + 4y + 5 = 0 is [2025]

1 $5 \sqrt{3}$

2 $15 \sqrt{5}$

3 12

4 15

Ans.
(4)

We have, $(x^{2} + 2 x) (12 - k) = 2$

$\Rightarrow λ x^{2} + 2 λ x - 2 = 0, k \neq 12$ [Let 12 – k = $λ$ ]

For equal roots, D = 0

$\Rightarrow 4 λ^{2} + 8 λ = 0$

$\Rightarrow 4 λ (λ + 2) = 0$

$\Rightarrow λ = 0$ (which is not possible)

or $λ = - 2 \Rightarrow 12 - k = - 2$

$\Rightarrow k = 14$

So, $P (k, \frac{k}{2}) = (14, 7)$

$∴ Distance, d =$ $| \frac{3 \times 14 + 4 \times 7 + 5}{\sqrt{9 + 16}} |$ $= 15$ .

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