Math Question #12. ICTM R 96 AA. Math Team.?

Question

Two circles (x + 4)^2 + (y - 7)^2 = 2 and (x - 3)^2 = 2 have common tangents. Find the equation of the common external tangent that has the largest y-intercept. Give your answer in the form y = mx + b.

Please show STEP BY STEP work.
Else this is pointless

Answer 1

You are missing the part of the equation for the second circle so there is no way to fully solve the problem, but here are some of the thinking behind the solution.

If the two circles don't not are overlap, there are four common tangents: two where both centers are on the same side of of the tangent line and where they are on opposite sides of the tangent line.

By "external" tangent, I assume you mean the ones where the centers are on the same side of the tangent line.

Now consider the quadrilateral formed by the two centers and the two tangent points of an external tangent. In general, you know the lengths of three sides (the two radii and the line between the two centers) and two angles (90 degrees at the tangents) so you have enough information to solve for the location of the tangent points.

But if the two circles have the same radius (as seems to be the case here), things are much simpler: the quadrilateral is a rectangle! So the angles at the centers are also 90 degrees so you can compute the tangent points in a straight-forward way.

Also, the slope of both the tangents is the same as the slope of the line between the centers so all you need from the tangent points is the offset.

In the more general case, when the radii are not equal, you can compute the angle at the center of the larger angle as follows.

Consider a right triangle with hypotenuse equal to the distance between the centers and one side equal to the difference in radii. The interior angle where these two sides meet is the angle at the larger center.

To see this:
let O and O' be the centers of the larger and smaller circles.
let T and T' be the corresponding tangent points.
let P be the point on the radius OT such that the length of PT is the same as that of O'T'

Since PT = O'T' and PTT' and TT'O' are both right angles, PTT'O' is a rectangle.

That means that both O'PT and O'PO are right angles.

So O'PO is a right triangle with OO' as its hypotenuse and OP have a length equal to the difference in the radii

So, whatever the full equation for the second circle looks like, you have a way of determining the answer.

Answer 2

i theory that the question grow to be asking for particular numbers to symbolize x and y. First, I talked approximately as the undemanding ratio "z". so as meaning: a^3/b * z^3 = b^3/a and z^3 = b^4/a^4 in case you replace some numbers in you will locate that what works suited is: z= sixteen b= 8 a=a million plug that throughout the time of and the series is; a million/8, 2, 32, 512 (x,y) = (2,32) in retrospect, i do no longer think of that's what the question asked for, yet ohh nicely.