Coordinates problem

[Guest]

Hey. I am trying to do this coordinate problem and I have no clue where to start or what to do. This is what I am up against:

Find the coordinates of a point in the third quadrant where the line y = 2x + 5 intersects the circle x^2 + y^2 = 25

a. (3, 4)
b. (4, 3)
c. (5, 2)
d. (4, -3)
e. (3, -4)
f. (-3, -4)
g. (-4, -3)
h. (-4, 3)
i. (-3, 4)
j. (0, -5)
k. (0, 5)
l. (5, 0)
m. (-5, 0)
n. none of these

Can someone help me figure this out?

 

[soroban]

Hello, achillesg22!

You don't have a clue??
. . You've never found the intersection of two graphs before?

Find the coordinates of a point in the third quadrant
where the line \(\displaystyle y\,=\,2x\,+\,5\) intersects the circle \(\displaystyle x^2\,+\,y^2\,=\,25\)

. . . [g] (-4, -3)

Solve the system of equations.

Substitute the equation of the line into the equation of the circle:

. . . . . . . . . .\(\displaystyle x^2\,+\,(2x + 5)^2\:=\:25\)

. . . \(\displaystyle x^2\,+\,4x^2\,+\,20x\,+\,25\:=\:0\)

. . . . . . . . . . . . \(\displaystyle 5x^2\,+\,20x\:=\:0\)

. . . . . . . . . . . . .\(\displaystyle 5x(x\,+\,4)\:=\:0\)

And we have: .\(\displaystyle x\,=\,0,-4\)

We also have: .\(\displaystyle y\,=\,5,\,-3\)


The points of intersection are: .\(\displaystyle (0,5),\-4,-3)\)

. . But \(\displaystyle (0,5)\) is not in the third quadrant . . . the answer is not [k].

Answer: .\(\displaystyle (-4,-3)\) . . . answer (g)