Question

Question #50f9b

Answer 1

`(a) k->y=(-4x+25)/3, l->y=(-x+10)/2; (b) D(4,3), (x-4)^2+(y-3)^2=50, r=5sqrt(2)`

Explanation:

Repeating the points
`A(5,10)`
`B(-3,4)`
`C(-1,-2)`
So the midpoints of interest are
`M_(AB)(1,7)`
`M_(AC)(2,4)`

(a)
Finding the slopes (`k_n=(Delta y)/(Delta x) and p_n=-1/(k_n)`)

`AB-> k_1=(4-10)/(-3-5)=(-6)/-8=3/4` => `p_1=-4/3`
`AC-> k_2=(-2-10)/(-1-5)=(-12)/(-6)=2` => `p_2=-1/2`

Equations of lines

`k-> (y-7)=(-4/3)(x-1)` => `y=(-4x+4)/3+7` => `y=(-4x+25)/3`
`l->(y-4)=(-1/2)(x-2)` => `y=(-x+2)/2+4` => `y=(-x+10)/2`

(b)
`D=knnl`

Finding D

`(-4x+25)/3=(-x+10)/2` => `-8x+50=-3x+30` => `5x=20` => `x=4`
`->y=(-4+10)/2=6/2` => `y=3`
`-> D(4,3)`

Since D is the circumcenter then
`AD=BD=CD=r`
(choosing CD)`=> r=sqrt((4+1)^2+(3+2)^2)=sqrt(25+25)=sqrt(50)=5sqrt(2)`

Circle's equation
`(x-x_D)^2+(y-y_D)^2=r^2`
`(x-4)^2+(y-3)^2=50`