Key Concepts and Formulas

• Quadratic Equation and Discriminant: For a quadratic equation $ax^2 + bx + c = 0$, the discriminant is $D = b^2 - 4ac$. The equation has equal roots if and only if $D = 0$.
• Distance of a Point from a Line: The distance $d$ from a point $(x_1, y_1)$ to a line $Ax + By + C = 0$ is given by $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$.

Step-by-Step Solution

Step 1: Transforming the equation into standard quadratic form and identifying coefficients

Our first goal is to rewrite the given equation $x(x+2)(12-k)=2$ in the standard quadratic form $ax^2 + bx + c = 0$ to apply the discriminant condition.

$x(x+2)(12-k) = 2$ $(x^2 + 2x)(12-k) = 2$ $(12-k)x^2 + 2(12-k)x = 2$ $(12-k)x^2 + 2(12-k)x - 2 = 0$

Now, we can identify the coefficients: $a = 12 - k$ $b = 2(12 - k)$ $c = -2$

Why this step? The discriminant formula $D = b^2 - 4ac$ is defined for a quadratic equation in its standard form. Correctly identifying $a$, $b$, and $c$ is crucial for its application.

Step 2: Applying the equal roots condition (D=0) to find k

For the quadratic equation to have equal roots, the discriminant must be zero. $D = b^2 - 4ac = 0$ $[2(12-k)]^2 - 4(12-k)(-2) = 0$ $4(12-k)^2 + 8(12-k) = 0$ $4(12-k)[(12-k) + 2] = 0$ $4(12-k)(14-k) = 0$

This gives us two possible values for $k$: $12 - k = 0 \Rightarrow k = 12$ $14 - k = 0 \Rightarrow k = 14$

If $k = 12$, the original equation becomes $x(x+2)(12-12) = 2$, which simplifies to $0 = 2$, a contradiction. Therefore, $k = 12$ is not a valid solution.

However, let's re-examine the original equation. If $k=12$, the equation is not quadratic, and the concept of "equal roots" does not apply. We divide by $12-k$ to obtain a new quadratic, which is only valid when $k \neq 12$.

Alternatively, suppose we proceed by dividing $(12-k)$ early: $x^2 + 2x = \frac{2}{12-k}$ $x^2 + 2x - \frac{2}{12-k} = 0$ Here, $a = 1$, $b = 2$, $c = -\frac{2}{12-k}$. Applying $D = b^2 - 4ac = 0$: $2^2 - 4(1)(-\frac{2}{12-k}) = 0$ $4 + \frac{8}{12-k} = 0$ $1 + \frac{2}{12-k} = 0$ $\frac{2}{12-k} = -1$ $2 = -12 + k$ $k = 14$

Why this approach? This approach highlights the importance of $12-k \neq 0$. If $12-k = 0$, i.e., $k=12$, the original equation becomes $x(x+2)(0) = 2$, which simplifies to $0 = 2$. This is a contradiction, meaning $k=12$ is not a valid solution that yields a quadratic equation with equal roots. Therefore, $k=14$ is the only valid value.

Thus, we have found $k=14$.

Step 3: Determining the coordinates of the point

The point is given as $(k, \frac{k}{2})$. Substitute $k=14$ into the coordinates: Point $(x_1, y_1) = (14, \frac{14}{2}) = (14, 7)$.

Why this step? The distance formula requires specific numerical coordinates for the point, which we can now obtain using the value of $k$ found in Step 2.

Step 4: Applying the point-to-line distance formula

The line equation is $3x + 4y + 5 = 0$. Comparing this to the standard form $Ax + By + C = 0$, we have: $A = 3$ $B = 4$ $C = 5$ The point is $(x_1, y_1) = (14, 7)$.

Now, substitute these values into the distance formula: $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$ $d = \frac{|3(14) + 4(7) + 5|}{\sqrt{3^2 + 4^2}}$

Why this step? This formula directly calculates the shortest (perpendicular) distance from the point to the line, which is what the problem asks for.

Step 5: Calculation and Final Answer

Numerator: $|3(14) + 4(7) + 5| = |42 + 28 + 5| = |75| = 75$

Denominator: $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Distance: $d = \frac{75}{5} = 15$

Thus, the distance of the point $(k, \frac{k}{2})$ from the line $3x+4y+5=0$ is $15$.

Common Mistakes & Tips

• Always ensure the quadratic equation is in the standard form before applying the discriminant formula.
• Be careful with terms that might make the coefficient of $x^2$ zero, as this changes the nature of the equation from quadratic to linear.
• Remember the absolute value in the numerator of the distance formula, as distance must be non-negative.

Summary

This problem combines the concepts of the discriminant of a quadratic equation and the distance from a point to a line. By setting the discriminant to zero, we found the value of $k$ to be 14. Subsequently, we calculated the distance of the point $(14, 7)$ from the line $3x + 4y + 5 = 0$, which resulted in a distance of 15.

The final answer is \boxed{15}, which corresponds to option (A).