Question:

Step 1: Identify the properties of the given circles. Circle $C_1$ has the equation $x^2 + y^2 = 16$. From this, we can identify its center $O_1$ and radius $R_1$. $$O_1 = (0,0)$$ $$R_1 = \sqrt{16} = 4$$ Circle $C_2$ has a radius $R_2 = 5$ units. Let its center be $O_2 = (h,k)$. Step 2: Determine the slope of the line connecting the centers. The common chord of two intersecting circles is always perpendicular to the line joining their centers. The problem states that the slope of the common chord is $m_c = \frac{3}{4}$. Therefore, the slope of the line $O_1O_2$, denoted $m_{O_1O_2}$, must be the negative reciprocal of $m_c$. $$m_{O_1O_2} = -\frac{1}{m_c} = -\frac{1}{3/4} = -\frac{4}{3}$$ Since $O_1 = (0,0)$ and $O_2 = (h,k)$, the slope of $O_1O_2$ is $\frac{k-0}{h-0} = \frac{k}{h}$. Equating the slopes, we get an equation relating $h$ and $k$: $$\frac{k}{h} = -\frac{4}{3} \implies 3k = -4h \implies 4h + 3k = 0 \quad (*)$$ Step 3: Interpret "common chord of maximum length". The length of the common chord of two intersecting circles is maximized when the chord passes through the center of the circle with the smaller radius. Comparing the radii, $R_1 = 4$ and $R_2 = 5$. Since $R_1 < R_2$, the common chord of maximum length must pass through the center of $C_1$, which is $O_1(0,0)$. Given that the common chord has a slope of $\frac{3}{4}$ and passes through $O_1(0,0)$, its equation is: $$y - 0 = \frac{3}{4}(x - 0) \implies y = \frac{3}{4}x \implies 3x - 4y = 0$$ Step 4: Calculate the distance from $O_2$ to the common chord. If the common chord passes through $O_1(0,0)$, its length as a chord of $C_1$ is $2R_1 = 2(4) = 8$. For this to be the common chord of both circles, its length as a chord of $C_2$ must also be $8$. Let $p_2$ be the perpendicular distance from the center $O_2(h,k)$ to the common chord $3x - 4y = 0$. The length of a chord in $C_2$ is given by $2\sqrt{R_2^2 - p_2^2}$. Setting this equal to $8$: $$2\sqrt{R_2^2 - p_2^2} = 8$$ $$\sqrt{5^2 - p_2^2} = 4$$ $$25 - p_2^2 = 16$$ $$p_2^2 = 9 \implies p_2 = 3$$ Now, we can express $p_2$ using the formula for the distance from a point $(h,k)$ to a line $Ax+By+C=0$: $$p_2 = \frac{|3h - 4k|}{\sqrt{3^2 + (-4)^2}} = \frac{|3h - 4k|}{5}$$ Equating this to $3$: $$\frac{|3h - 4k|}{5} = 3 \implies |3h - 4k| = 15$$ This gives two possibilities: $$3h - 4k = 15 \quad (**)$$ or $$3h - 4k = -15 \quad (***)$$ Step 5: Solve the system of equations for the coordinates $(h,k)$. We have two cases based on equations $(**)$ and $(***)$ combined with equation $(*)$: From $(*)$, we have $k = -\frac{4}{3}h$. Case 1: Using equation $(**)$, $3h - 4k = 15$. Substitute $k = -\frac{4}{3}h$: $$3h - 4\left(-\frac{4}{3}h\right) = 15$$ $$3h + \frac{16}{3}h = 15$$ $$\frac{9h + 16h}{3} = 15$$ $$\frac{25h}{3} = 15$$ $$25h = 45 \implies h = \frac{45}{25} = \frac{9}{5}$$ Now find $k$: $$k = -\frac{4}{3}h = -\frac{4}{3} \cdot \frac{9}{5} = -\frac{12}{5}$$ So, one possible center for $C_2$ is $\left(\frac{9}{5}, -\frac{12}{5}\right)$. Case 2: Using equation $(***)$, $3h - 4k = -15$. Substitute $k = -\frac{4}{3}h$: $$3h - 4\left(-\frac{4}{3}h\right) = -15$$ $$3h + \frac{16}{3}h = -15$$ $$\frac{25h}{3} = -15$$ $$25h = -45 \implies h = -\frac{45}{25} = -\frac{9}{5}$$ Now find $k$: $$k = -\frac{4}{3}h = -\frac{4}{3} \cdot \left(-\frac{9}{5}\right) = \frac{12}{5}$$ So, another possible center for $C_2$ is $\left(-\frac{9}{5}, \frac{12}{5}\right)$. Step 6: State the final answer. The coordinates of the center of $C_2$ are $\left(\frac{9}{5}, -\frac{12}{5}\right)$ or $\left(-\frac{9}{5}, \frac{12}{5}\right)$. These can be collectively represented as $\left(\pm\frac{9}{5}; \mp\frac{12}{5}\right)$ (where the signs are opposite). This matches the format of Option 2. The final answer is $\boxed{\left(\pm\frac{9}{5}; \pm\frac{12}{5}\right)}$