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
(b) intersects the graph at exactly 10 points
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?
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]
Thank you for helping me!