Question

Question #3ddb7

Answer 1

`( (20sqrt17)/17, (-5sqrt17)/17),( (-20sqrt17)/17, (5sqrt17)/17)`

Explanation:

finding the equation of the circle:

the general equation of a circle centered at the origin is `x^2+y^2=r^2`,
where `r` is the radius of the circle.

if the radius is `5`, `r^2 = 25`.

the equation of the circle is therefore `x^2+y^2=25`.

finding points of intersection:

solve simultaneous equations `x^2+y^2=25` and `y=-x/4`.

(here, it is easier to insert `y` than `x`)

`x^2+y^2=25`
`x^2+(-x/4)^2=25`

`(-x/4)^2 = x^2/16`

`x^2+x^2/16 =25`

`(16x^2)/16 + x^2/16 = 25`

`(17x^2)/16 = 25`

`17x^2 = 25 * 16 = 400`

`x^2 = 400/17`

`x = sqrt(400/17) = 20/sqrt17` or `-20/sqrt17`

`x = (20sqrt17)/17` or `(-20sqrt17)/17`

`y = -x/4 = (-5sqrt17)/17` or `(5sqrt17)/17`

coordinates: `( (20sqrt17)/17, (-5sqrt17)/17),( (-20sqrt17)/17, (5sqrt17)/17)`

Answer 2

`P_1: (4.85, -1.21)`
`P_2: (-4.85, 1.21)`

Explanation:

Set radius = r = 5

The equation of the circle:

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

`x^2 + y^2 = 25`

The equation of the line:

`y = -x/4`

Insert equation of the circle in the equation of the line to find the intersection points.

`x^2 + y^2 = 25`

`x^2 + (-x/4)^2 = 25`

`x^2 + (x^2/16) = 25`

`(16x^2)/17 = 25`

`x^2 = (25*17)/16`

`x = +- sqrt((25*17)/16)`

`x = +- 4.85`

So, the x values are:

`x_1 = 4.85`

`x_2 = -4.85`

NOTE: It is important to always put the`+-` in front of a square root, because both the negative and the positive answer can be correct.

Now, find the y value by plugging the two x values in any of the two equations.

`y_1 = -(-4.85)/4 = 1.21`

`y_2 = -(4.85)/4 = -1.21`