Equal Roots Condition for the Equation
The given equation is $x(x + 2)(12-k) = 2$.
Expand the equation:
$(x^2 + 2x)(12-k) = 2$
$(12-k)x^2 + 2(12-k)x - 2 = 0$
This is a quadratic equation of the form $Ax^2 + Bx + C = 0$, where:
- $A = 12-k$
- $B = 2(12-k)$
- $C = -2$
For the equation to have equal roots, the discriminant ($\Delta$) must be zero.
$\Delta = B^2 - 4AC = 0$
Substitute the values of A, B, and C:
$(2(12-k))^2 - 4(12-k)(-2) = 0$
$4(12-k)^2 + 8(12-k) = 0$
Factor out $4(12-k)$:
$4(12-k)[(12-k) + 2] = 0$
$4(12-k)(14-k) = 0$
This implies either $12-k = 0$ or $14-k = 0$.
Therefore, $k=12$ or $k=14$.
If $k=12$, the original equation becomes $x(x+2)(0) = 2$, which simplifies to $0=2$. This is impossible.
Thus, we must have $k=14$.
Distance Calculation from Point to Line
The point is $(k, \frac{k}{2})$. Substituting $k=14$, the point is $(14, \frac{14}{2}) = (14, 7)$.
The line is given by $3x + 4y + 5 = 0$. Here, $A=3$, $B=4$, $C=5$.
The formula for the distance ($d$) from a point $(x_0, y_0)$ to a line $Ax + By + C = 0$ is:
$d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$
Substitute the coordinates of the point $(14, 7)$ and the line coefficients:
$d = \frac{|3(14) + 4(7) + 5|}{\sqrt{3^2 + 4^2}}$
$d = \frac{|42 + 28 + 5|}{\sqrt{9 + 16}}$
$d = \frac{|75|}{\sqrt{25}}$
$d = \frac{75}{5}$
$d = 15$
The distance is 15.
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