Key Concepts and Formulas

• Equation of a Circle: The general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$. Its center is $C = (-g, -f)$ and its radius is $r = \sqrt{g^2 + f^2 - c}$.
• Equation of the Chord of Contact: If tangents are drawn from an external point $P(x_1, y_1)$ to a circle $S \equiv x^2 + y^2 + 2gx + 2fy + c = 0$, the equation of the chord joining the points of contact (chord of contact) is given by $T=0$, where $T = xx_1 + yy_1 + g(x+x_1) + f(y+y_1) + c$.
• Perpendicular Distance from a Point to a Line: The perpendicular distance from a point $(x_0, y_0)$ to a line $Ax + By + C = 0$ is given by $d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$.
• Pythagorean Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Step-by-Step Solution

Step 1: Identify the Circle's Properties and the External Point

The given equation of the circle is $x^2 + y^2 - 8x - 4y + 16 = 0$. We need to find the center and radius of the circle. Comparing this with the general equation $x^2 + y^2 + 2gx + 2fy + c = 0$, we have: $2g = -8 \Rightarrow g = -4$ $2f = -4 \Rightarrow f = -2$ $c = 16$

• Center of the circle (C): $C = (-g, -f) = (4, 2)$.
• Radius of the circle (r): $r = \sqrt{g^2 + f^2 - c} = \sqrt{(-4)^2 + (-2)^2 - 16} = \sqrt{16 + 4 - 16} = \sqrt{4} = 2$ units.

The tangents are drawn from the origin, so the external point $P(x_1, y_1) = (0, 0)$.

Step 2: Find the Equation of the Chord of Contact (AB)

The points A and B are the points of tangency. The line segment AB is the chord of contact. Using the formula $T=0$ for the chord of contact from $P(0,0)$ to the circle $x^2 + y^2 - 8x - 4y + 16 = 0$: $xx_1 + yy_1 + g(x+x_1) + f(y+y_1) + c = 0$ Substituting $x_1=0$, $y_1=0$, $g=-4$, $f=-2$, $c=16$: $x(0) + y(0) + (-4)(x+0) + (-2)(y+0) + 16 = 0$ $0 + 0 - 4x - 2y + 16 = 0$ $-4x - 2y + 16 = 0$ Dividing by -2, we get the equation of the chord of contact AB: $2x + y - 8 = 0$

Step 3: Calculate the Perpendicular Distance from the Center to the Chord of Contact (CM)

Let M be the midpoint of the chord AB. The line segment CM is perpendicular to AB. We need to find the distance from the center $C(4,2)$ to the line (chord of contact) $2x + y - 8 = 0$. Using the perpendicular distance formula $d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$: Here, $(x_0, y_0) = (4,2)$ and the line is $2x + 1y - 8 = 0$ (so $A=2, B=1, C=-8$). $CM = \frac{|2(4) + 1(2) - 8|}{\sqrt{2^2 + 1^2}} = \frac{|8 + 2 - 8|}{\sqrt{4 + 1}} = \frac{|2|}{\sqrt{5}} = \frac{2}{\sqrt{5}} \text{ units}$

Step 4: Calculate the Half-Length of the Chord of Contact (AM)

Consider the right-angled triangle CMA, where M is the midpoint of AB.

• CA is the radius of the circle, so $CA = r = 2$.
• CM is the perpendicular distance from the center to the chord, which we just calculated as $2/\sqrt{5}$.
• AM is half the length of the chord AB. Applying the Pythagorean theorem ($CA^2 = CM^2 + AM^2$): $2^2 = \left(\frac{2}{\sqrt{5}}\right)^2 + AM^2$ $4 = \frac{4}{5} + AM^2$ $AM^2 = 4 - \frac{4}{5} = \frac{20 - 4}{5} = \frac{16}{5}$ $AM = \sqrt{\frac{16}{5}} = \frac{4}{\sqrt{5}} \text{ units}$

Step 5: Calculate the Length of the Chord of Contact (AB)

Since M is the midpoint of AB, the length of the chord AB is twice the length of AM. $AB = 2 \times AM = 2 \times \frac{4}{\sqrt{5}} = \frac{8}{\sqrt{5}} \text{ units}$

Step 6: Calculate $(AB)^2$

We need to find the square of the length of the chord AB. $(AB)^2 = \left(\frac{8}{\sqrt{5}}\right)^2 = \frac{64}{5}$

Common Mistakes & Tips

• Be careful with signs when finding the center of the circle from its equation.
• Remember to square the entire fraction when calculating $(AB)^2$.
• Drawing a diagram is always helpful in circle geometry problems.

Summary

We found the center and radius of the given circle. Then, we determined the equation of the chord of contact using the formula $T=0$. Next, we calculated the perpendicular distance from the center to the chord. Using the Pythagorean theorem, we found half the length of the chord and then the full length. Finally, we squared the length of the chord to obtain the desired result.

Final Answer

The final answer is \boxed{\frac{64}{5}}, which corresponds to option (D).