SOLUTION: find all positive real number m so that the graphs of x+my=0 and x=y^2 have respectively: a. exactly one point of intersection b. no point of intersection c. exactly 2 points of

Algebra ->  Coordinate-system -> SOLUTION: find all positive real number m so that the graphs of x+my=0 and x=y^2 have respectively: a. exactly one point of intersection b. no point of intersection c. exactly 2 points of      Log On


   

Question 729832: find all positive real number m so that the graphs of x+my=0 and x=y^2 have respectively:
a. exactly one point of intersection
b. no point of intersection
c. exactly 2 points of intersection.
Answer by hoomanc(14) About Me  (Show Source):
You can put this solution on YOUR website!
first re-arrange the equation base on x:
y=%28-1%2Fm%29%2Ax and y=%2Bsqrt%28x%29 or y=-sqrt%28x%29 by writing the equilibrium for both sides base on "y" we would have:
%28-1%2Fm%29%2Ax=sqrt%28x%29 then %281%2Fm%5E2%29%2Ax%5E2=x then %281%2Fm%5E2%29%2Ax%5E2-x=0 then
x%5E2-%28m%5E2%29%2Ax=0 then x%2A%28X-m%5E2%29=0 that means:
X=0 or X=m%5E2 so, base on this answer it appears that regardless of what value m taken we always would have an intersection in the origin of x=0 & y=0.
In addition, if we try to use a very small value for m, lets say for example 1%2F10%5E100000000 the second answer of x would be very close value to zero , In other words, as much as we take a smaller value for m we are more closer to a unique intersection value of zero, otherwise in bigger cases we can't neglect bigger values of m and the second intersection would be at x=m%5E2.

In above graph the two curve line are y=sqrt%28x%29 and y=-sqrt%28x%29
As much as the take a smaller value for m the straight line is going to get more vertical {(red m=1) (green m=0.2) (blue m= 0.1)}. so the second intersection point get closer more and more to zero.


In conclusion >>>
1)for case b: we can't guarantee that for sure cause there will always be at least one intersection at X=0
2) for case a if we try to minimize the value of m we are close to unique answer of one intersect at X=0 but it's never gonna happen (try to draw both graphs to understand why)
3)for case c: it's what is always going to happen for the rest of non-small values of m.
hope that helps.