Find the total length of 2 circles and 2 tangents

The following question has this supporting diagram .

An ear-ring is made from silver wire and is designed in the shape of 2 touching circles with two tangents to the outer circle, as show in diagram 1.

Diagram 2 shows a drawing of this ear-ring related to the coordinate axes. The circles touch at (0, 0). The equation of the inner circle is ${x^2 + y^2 + 3y = 0}$. The outer circle intersects the y-axis (0, -4). The tangents meet the y-axis at (0, -6). Find the total length of silver wire required to make the ear-ring.

My take away from this is that I need to find the total of the circumference of both circles and the length of both tangents.

I can get the centre of the small circle from the equation of the small circle -2g as ${- 3\over 2}$, I can then say the radius is ${0 - -{3\over 2}}$ or ${3\over 2}$. I can then use ${2 \pi r}$ to say the circumference of the small circle which is roughly 9.45.

I can work out the radius of the large circle by find the distance between (0, 0) and (0, -4) and dividing by 2 which is 2 so the circumference of the big circle which is approximately 12.5.

For the 2 tangents I know the lengths of 2 of the sides because the tangent and the radius form a right angle and the radius of the large circle is 2 and I know the distance from the start of the tangent (0, -6) to the centre of the large circle which is (0, -2) and a length of 4.

I can use Pythagoras's theorem to work out the tangent length.

${a^2 + 2^2 = 4^2}$

=> ${a = {\sqrt 12}}$

Multplied by 2 is roughly 7

So my total is the sum of

small circle = 9.45 large cicle = 12.5 2 tangents = 7

My answer is then 28.95

Am I on the right track or have I gone mad?

1 Answer

Yes. From the equations and the drawing given, it is evident that the circles have diameters $3$ and $4$, so their circumferences are $3\pi$ and $4\pi$, respectively, adding up to $7\pi$.

To this must be added the lengths of the tangents. You can determine them as you have. Alternatively, you can use the fact that a tangent to a circle from a point $P$ has length equal to the square root of the product of two lengths: $P$ to the nearest point on the circle, and $P$ to the furthest point on the circle.

In this case, those two lengths are $2$ and $6$, so each tangent has length $\sqrt{2 \cdot 6} = \sqrt{12} = 2\sqrt{3}$. Therefore, the total length is $7\pi+4\sqrt{3} \doteq 28.92$.

• So I guess your sanity is safe, for now. :-) – Brian Tung Sep 30 '15 at 20:04
• yes but it is early :) – dagda1 Sep 30 '15 at 20:06
• BTW, I fixed a small typo in the final sentence: I had previously written $7\pi + 2\sqrt{3}$; the correct expression is $7\pi+4\sqrt{3}$. The decimal approximation was appropriate and remains unchanged. – Brian Tung Sep 30 '15 at 20:14