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**Tangera** Hello! I wonder if there's a better way of solving the questions below rather than sketching out the curves and circles or substiting in values of r and checking by trial and error. Please give suggestions. Thank you! :)

1) Consider the graph described by the equation y = tan x. The equation of a circle centred at the origin may also be written as

y = __+__ (r^2 - x^2)^(1/2)

after rearrangement. Give an equation of a circle centred at the origin that

(a) intersects the graph at exactly 2 points

--> Based on my understanding of the tangent graph,

y = __+__ (pi^2 - x^2)^(1/2) fulfils the requirement :) Mr F says: That won't work. Play around with $\displaystyle 0 < r < \frac{\pi}{2}$ ....

(b) intersects the graph at exactly 10 points Mr F says: Play around with $\displaystyle \frac{3 \pi}{2} < r < \frac{5 \pi}{2}$ ....

2) Is there a circle centred at the origin that interesects the graph at exactly 4 points? If so, what is its equation? If not, why not? Mr F says: Nice question .... as I played around with some rough graphs it certainly looks possible ...... You obviously need the circle to be TANGENT (touching but not crossing) to the parts of the tan graph between

$\displaystyle \frac{\pi}{2}$ and $\displaystyle \frac{3\pi}{2}$,

and

$\displaystyle \frac{-\pi}{2}$ and $\displaystyle \frac{-3\pi}{2}$ .....

And, as TKH pointed out "if you [touch] y = tan(x) at (a,b), you MUST [touch] it again at (-a,-b)" .... If you think about it, you'll see that the radius of the circle will $\displaystyle \sqrt{a^2 + b^2}$, so the question boils down to getting a and b ..... See my main reply below.

3) Can a circle centred at the origin intersect the graph at an __odd__ number of points?

--> I don't think so, because when the circle cuts the graph at one point,say, point A, it will also cut another point on the graph that is symmetrical to point A about the origin. [Nature of the y = tan x graph] Mr F says: You're right. It gets back to what TKH said about (a, b) and (-a, b) ........ You will always have PAIRS of intersection points => only an even number.

Thank you for helping me!