here are the problems:
1. The graphs of these two equations intersect in how many points?
x^2 + y^2 =25
x^2-y^2=3
2. For every value of a, the graph of each of these three equations is a line:
ax+y=1
x+ay=1
x+y=6
For what value of a do all three of these lines intersect at the same point?
3. A and B are constants such that the graphs of xy=B and y=Ax^2 intersect at the point (2,6).
Write the ordered pair (A,B).
thanks!
1. The graphs of these two equations intersect in how many points?
x^2 + y^2 =25
x^2- y^2 = 3
Add these equations and we get that
2x^2 = 28 divide through by 2
x^2 = 14 take both roots
x = sqrt (14) or x = -sqrt (14)
When x = sqrt (14) we have
(sqrt 14)^2 + y^2 = 25
14 + y^2 = 25
y^2 = 25 - 14
y^2 = 11
y = sqrt (11) or y = -sqrt (11)
So.......two points of intersection are (sqrt 14, sqrt 11) and ( sqrt 14, - sqrt 11)
Due to symmetry......the graphs will also intersect at (-sqrt 14, sqrt 11) and (-sqrt 14 , - sqrt 11)
So.....4 points of intersection
See the graph here : https://www.desmos.com/calculator/ie0t00e5nl
2. For every value of a, the graph of each of these three equations is a line:
ax+y=1
x+ay=1
x+y=6
For what value of a do all three of these lines intersect at the same point?
Note that the last equations can be writtern as y = 6 - x
Subbing this into the other two equations we get
ax + (6 - x) = 1 ⇒ x(a - 1) = -5
x + a(6 - x) = 1 ⇒ x((1 - a) = 1 -6a ⇒ x ( a - 1) = 6a - 1
This implies that 6a - 1 = -5 ⇒ 6a = -4 ⇒ a = -4/6 = -2/3
Subbing into the first equation for a and y we have that
(-2/3)x + 6 - x = 1
-(5/3)x + 6 = 1
-(5/3)x = -5
x = 3
And y =6- x = 6 - 3 = 3
The intersection pt is (3,3)
See the grpah here : https://www.desmos.com/calculator/nmla6yjdjt