Question

Find the points where the line y=−x/4 intersects a circle of radius 2 centered at the origin. Give exact values for the x and y coordinates.

Answer 1

Given

`y=-(x)/(4)`

We know that the general equation for a circle is given by:

`(x-h)^2+(x-k)^2=r^2`

Since the circle for this case is centered at the origin then h,k =0 and we have this:

`x^2+y^2=2^2`

For this case we can replace the formula for y from the line into equation (1) and we got:

`\begin{gathered} x^2+(-(x)/(4))^2=4 \\ \\ x^2+(x^2)/(16)=4 \\ multiply\text{ all through by 16} \\ 16x^2+x^2=64 \\ 17x^2=64 \\ \end{gathered}`

Divide both sides by 17

`\begin{gathered} (17x^2)/(17)=(64)/(17) \\ \\ x^2=(64)/(17) \\ \\ \\ Take\text{ the square root} \\ x=\sqrt{(64)/(17)} \\ \\ x=(8)/(√(17)) \\ Rationalize \\ \\ x=\pm(8√(17))/(17) \end{gathered}`

Now we just need to replace into the original equation for the line and we get the y coordinates like this:

`\begin{gathered} y=-(((8√(17))/(17)))/(4) \\ \\ y=-(2√(17))/(17) \end{gathered}`

and

`\begin{gathered} y=-(((-8√(17))/(17)))/(17) \\ \\ y=(8√(17))/(289) \end{gathered}`

The final answer

`((8√(17))/(17),-(2√(17))/(17)),((8√(17))/(17),(8√(17))/(289))`