Key Concepts and Formulas

• Equation of a Circle Touching the Y-axis: A circle touching the y-axis at $(0, k)$ has an equation of the form $(x-h)^2 + (y-k)^2 = h^2$, where $(h, k)$ is the center and $|h|$ is the radius. Alternatively, we can express this as $x^2 + y^2 - 2hx - 2ky + k^2 = 0$.
• Distance from a Point to a Line: The perpendicular distance from a point $(x_1, y_1)$ to a line $ax + by + c = 0$ is given by $d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}$.
• Tangent Condition: A line is tangent to a circle if and only if the perpendicular distance from the center of the circle to the line is equal to the radius of the circle.

Step-by-Step Solution

Step 1: Find the equation of the circle.

Since the circle touches the y-axis at (0, 4), the center of the circle has coordinates (h, 4), and the radius is |h|. Thus, the equation of the circle is $(x - h)^2 + (y - 4)^2 = h^2$. Since the circle passes through the point (2, 0), we can substitute x = 2 and y = 0 into the equation: $(2 - h)^2 + (0 - 4)^2 = h^2$ $4 - 4h + h^2 + 16 = h^2$ $20 - 4h = 0$ $4h = 20$ $h = 5$

So, the equation of the circle is $(x - 5)^2 + (y - 4)^2 = 5^2$, which simplifies to $x^2 - 10x + 25 + y^2 - 8y + 16 = 25$, or $x^2 + y^2 - 10x - 8y + 16 = 0$. The center of the circle is (5, 4) and the radius is 5.

Step 2: Check which line is NOT a tangent to the circle.

A line is tangent to the circle if the distance from the center of the circle to the line equals the radius. We will calculate the distance from the center (5, 4) to each line and check if it equals 5.

(A) 3x - 4y - 24 = 0 $d = \frac{|3(5) - 4(4) - 24|}{\sqrt{3^2 + (-4)^2}} = \frac{|15 - 16 - 24|}{\sqrt{9 + 16}} = \frac{|-25|}{5} = 5$

(B) 4x + 3y - 8 = 0 $d = \frac{|4(5) + 3(4) - 8|}{\sqrt{4^2 + 3^2}} = \frac{|20 + 12 - 8|}{\sqrt{16 + 9}} = \frac{|24|}{5} = \frac{24}{5} \neq 5$

(C) 3x + 4y - 6 = 0 $d = \frac{|3(5) + 4(4) - 6|}{\sqrt{3^2 + 4^2}} = \frac{|15 + 16 - 6|}{\sqrt{9 + 16}} = \frac{|25|}{5} = 5$

(D) 4x - 3y + 17 = 0 $d = \frac{|4(5) - 3(4) + 17|}{\sqrt{4^2 + (-3)^2}} = \frac{|20 - 12 + 17|}{\sqrt{16 + 9}} = \frac{|25|}{5} = 5$

Since the distance from the center to the line 4x + 3y - 8 = 0 is not equal to the radius, this line is not a tangent to the circle.

Common Mistakes & Tips

• Be careful with the signs when calculating the distance from a point to a line.
• Remember the correct formula for the equation of a circle touching the y-axis.
• Double-check your arithmetic to avoid errors.

Summary

We first found the equation of the circle using the given conditions. Then, we calculated the distance from the center of the circle to each of the given lines. The line for which this distance is not equal to the radius of the circle is not a tangent. The line 4x + 3y - 8 = 0 is not a tangent.

Final Answer

The final answer is \boxed{3x – 4y – 24 = 0}, which corresponds to option (A).