Key Concepts and Formulas

• The standard equation of a circle with center $(h, k)$ and radius $r$ is $(x-h)^2 + (y-k)^2 = r^2$.
• The perpendicular distance $d$ from a point $(x_1, y_1)$ to a line $Ax + By + C = 0$ is given by $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$.
• A line intersects a circle at two distinct points if the perpendicular distance from the center of the circle to the line is less than the radius of the circle.

Step-by-Step Solution

Step 1: Find the Center and Radius of the Circle

The given equation of the circle is $x^2 + y^2 - 2x - 4y + 4 = 0$. We need to rewrite this equation in the standard form $(x-h)^2 + (y-k)^2 = r^2$ to identify the center $(h, k)$ and radius $r$. We do this by completing the square.

Rearrange the terms: $(x^2 - 2x) + (y^2 - 4y) + 4 = 0$ Complete the square for the $x$ terms: $x^2 - 2x + 1 = (x - 1)^2$. Complete the square for the $y$ terms: $y^2 - 4y + 4 = (y - 2)^2$. Rewrite the equation: $(x^2 - 2x + 1) + (y^2 - 4y + 4) + 4 - 1 - 4 = 0$ $(x - 1)^2 + (y - 2)^2 - 1 = 0$ $(x - 1)^2 + (y - 2)^2 = 1$ Now, we can identify the center and radius:

• Center: $(h, k) = (1, 2)$
• Radius: $r = \sqrt{1} = 1$

Step 2: Express the Line Equation in the General Form

The equation of the line is $3x + 4y = k$. We need to rewrite it in the general form $Ax + By + C = 0$ for use in the distance formula. $3x + 4y - k = 0$ Here, $A = 3$, $B = 4$, and $C = -k$.

Step 3: Calculate the Distance from the Center of the Circle to the Line

We use the distance formula to find the perpendicular distance $d$ from the center of the circle $(1, 2)$ to the line $3x + 4y - k = 0$: $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$ Substitute the values: $d = \frac{|3(1) + 4(2) - k|}{\sqrt{3^2 + 4^2}}$ $d = \frac{|3 + 8 - k|}{\sqrt{9 + 16}}$ $d = \frac{|11 - k|}{\sqrt{25}}$ $d = \frac{|11 - k|}{5}$

Step 4: Apply the Condition for Two Distinct Intersection Points

For the line to intersect the circle at two distinct points, the distance $d$ must be less than the radius $r$: $d < r$ $\frac{|11 - k|}{5} < 1$ Multiply both sides by 5: $|11 - k| < 5$ This inequality is equivalent to: $-5 < 11 - k < 5$ Subtract 11 from all parts: $-5 - 11 < -k < 5 - 11$ $-16 < -k < -6$ Multiply all parts by -1 and reverse the inequality signs: $16 > k > 6$ $6 < k < 16$

Step 5: Determine the Number of Integral Values of k

We need to find the number of integers $k$ such that $6 < k < 16$. The integers are 7, 8, 9, 10, 11, 12, 13, 14, and 15. There are 9 integers in this range.

Common Mistakes & Tips

• When completing the square, double-check that you're adding the correct constant to both sides of the equation.
• Remember to reverse the inequality signs when multiplying or dividing by a negative number.
• Be careful when counting integers in an interval. Since the inequality is strict ($6 < k < 16$), the endpoints 6 and 16 are not included.

Summary

We found the center and radius of the circle, calculated the distance from the center to the line, and used the condition $d < r$ to find the range of values for $k$. The number of integral values of $k$ that satisfy the inequality is 9.

The final answer is \boxed{9}.