Question

the equation of a curve is xy=12 and the equation of a line l is 2x+y=k where k is a constant .In the case where k=11 find the coordinates of the points of intersection of l and the curve.

Answer 1

Equations are xy=12 and 2x+y=11

solve one for either x or y, and plug it into the other one to get coord.  I'll solve for x.

I first solve the problem on the left for y, and plug it into the second one, giving me only one variable to solve for.  solving left for y is y=12/x

Plug new y value into second equation and solve for x.
2x +12/x=11
mult both sides by x to clear denominator.  You get...
2x^2-11x+12=0
now factor and get (2x-3)(x-4)
Make them both equal to zero and solve for x.  
You get x=3/2 and x=4
Plug these into your original xy=12 equation to get the corresponding y coordinate for that specific x value.  You get 
(3/2, 8) and (4,3).  Your lines intersect twice, at these two points.