Lines and circles.

[DanDan]

Ok, here's my question and how far I have got so far.

Show that the line 2x+x=5 intersects the circle x²+y²=25 and find the coordinates of the point of intersection.

so I substitute x=-2x+5 into x²+y²=25

(-2y+5)²+y²=25
5y²-20y=0

next I put it into the discriminant which gives

20²-4x5x0=400

which makes 400 larger than 0 so I know there must be two points of intersection but I have no idea what to do to find what the points are. Please help if you can

 

[Mrspi]

Ok, here's my question and how far I have got so far.

Show that the line 2x+x=5 intersects the circle x²+y²=25 and find the coordinates of the point of intersection.

so I substitute x=-2x+5 into x²+y²=25

(-2y+5)²+y²=25
5y²-20y=0

next I put it into the discriminant which gives

20²-4x5x0=400

which makes 400 larger than 0 so I know there must be two points of intersection but I have no idea what to do to find what the points are. Please help if you can

PLEASE double-check to make sure that you've typed the problem correctly in the first place.

I'm very suspicious that "Show that the line 2x+x=5" is NOT correct, and that i probably should be "the line 2x + y = 5". I'm not going to waste any effort in trying to assist with this problem until I'm positive about what the problem actually says......

 

[DanDan]

My mistake

The problem should be 2y+5=x. Hope you can help now

 

[pka]

The problem should be 2y+5=x. Hope you can help now

Solve \(\displaystyle (2y+5)^2+y^2=25\) or \(\displaystyle 5y^2+20y=0\) for \(\displaystyle y\).

Then use those values to find the corresponding values for \(\displaystyle x\).

 

[DanDan]

I'm so sorry...

I typed it wrong AGAIN!

It is 2y+x=5. Which I have re-arranged to make x=-2y+5.

All the working up until the part where i use the discriminant is correct, I think.

Please excuse my lack of concentration when writing this post, I've been awake far too long but I just need to finish this exercise off and then sleep.

 

[pappus]

Ok, here's my question and how far I have got so far.

Show that the line 2x+x=5 intersects the circle x²+y²=25 and find the coordinates of the point of intersection.

so I substitute x=-2x+5 into x²+y²=25

(-2y+5)²+y²=25
5y²-20y=0 <--- OK

...

I typed it wrong AGAIN!

It is 2y+x=5. Which I have re-arranged to make x=-2y+5.

All the working up until the part where i use the discriminant is correct, I think.

...

\(\displaystyle 5y^2-20y=0~\implies~5y(y-4)=0\) Solve for y.

Plug in these results into the equation of the line to determine the corresponding x-values.

Btw a sketch could be very helpful to see which step has to be taken next.

 

[DanDan]

Yeah thanks for that. So when y=0 x=-2*0+5 = 5
and when y=4 x=-2*4+5 = -3

So my answers are (5,0) and (-3,4)

And I just checked the answers in the textbook and its all correct. Thanks a lot for your help