SOLUTION: (a) Given two equations of conic sections as follows, (x-1)(x+5)+(y+1)(y+5)=3, (x+1)(x-5)+8(y-5)=-17. (i) Find the vertex, focus and directrix of the parabola

Question 792247: (a) Given two equations of conic sections as follows,
(x-1)(x+5)+(y+1)(y+5)=3,
(x+1)(x-5)+8(y-5)=-17.

(i) Find the vertex, focus and directrix of the parabola
(ii)Find the center and radius of the circle.
(b) Find the shortest distance from the (-2,3) to the line 4x=3y + 2.
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
Given two equations of conic sections as follows,
(x-1)(x+5)+(y+1)(y+5)=3
x^2+4x-5 + y^2 + 6y +5 = 3
x^2+4x + ? + y^2+6y + ?? = 3
x^2+4x+4 + y^2+6y+9 = 3+4+9
(x+2)^2 + (y+3)^2 = 16
center is (-2,-3)
radius is 4
---------------------------
(x+1)(x-5)+8(y-5)=-17
x^2 -4x -5 + 8(y-5) = -17
x^2-4x+4 -5 +8(y-5) = -17 + 4
(x-2)^2 +8(y-5) = -8
(x-2)^2 = -8[(y-5)+1]
(x-2)^2 = -8(y-4)
----
p = -2
vertex:: (2,4)
focus: (2,2)
directrix: y = 6
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Cheers,
Stan H.
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(i) Find the vertex, focus and directrix of the parabola
(ii)Find the center and radius of the circle.
(b) Find the shortest distance from the (-2,3) to the line 4x=3y + 2.

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