document.write( "Question 729832: find all positive real number m so that the graphs of x+my=0 and x=y^2 have respectively:
\n" ); document.write( "a. exactly one point of intersection
\n" ); document.write( "b. no point of intersection
\n" ); document.write( "c. exactly 2 points of intersection. \n" ); document.write( "

Algebra.Com's Answer #446259 by hoomanc(14) $\"\"$ $\"About$

You can put this solution on YOUR website!
first re-arrange the equation base on $\"x\"$ :
\n" ); document.write( " $\"y=%28-1%2Fm%29%2Ax\"$ and $\"y=%2Bsqrt%28x%29\"$ or $\"y=-sqrt%28x%29\"$ by writing the equilibrium for both sides base on \"y\" we would have:
\n" ); document.write( " $\"%28-1%2Fm%29%2Ax=sqrt%28x%29\"$ then $\"%281%2Fm%5E2%29%2Ax%5E2=x\"$ then $\"%281%2Fm%5E2%29%2Ax%5E2-x=0\"$ then
\n" ); document.write( " $\"x%5E2-%28m%5E2%29%2Ax=0\"$ then $\"x%2A%28X-m%5E2%29=0\"$ that means:
\n" ); document.write( " $\"X=0\"$ 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 $\"x=0\"$ & $\"y=0\"$ .
\n" ); document.write( "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 $\"x=m%5E2\"$ .\r
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\n" ); document.write( "In above graph the two curve line are $\"y=sqrt%28x%29\"$ and $\"y=-sqrt%28x%29\"$
\n" ); document.write( "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.\r
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\n" ); document.write( "\n" ); document.write( "In conclusion >>>
\n" ); document.write( "1)for case b: we can't guarantee that for sure cause there will always be at least one intersection at $\"X=0\"$
\n" ); document.write( "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)
\n" ); document.write( "3)for case c: it's what is always going to happen for the rest of non-small values of $\"m\"$ .\r
\n" ); document.write( "\n" ); document.write( "hope that helps. \n" ); document.write( "

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