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What is the equation of the parabola with a focus at (8,2) and a directrix of y= 5?

Answer 1

Kennedy Adams

The equation is #(x-8)^2=-3(2y-7)#

Every point on the parabola is equally spaced from the directrix and the focus.

Consequently,

#sqrt((x-8)+(y-2))=5-y#

Squaring,

#(x-8)^2+(y-2)^2=(5-y)^2#

#(x-8)^2+cancely^2-4y+4=25-10y+cancely^2#

#(x-8)^2=-6y+21#

#(x-8)^2=-3(2y-7)#

graph{(x-8)^2+3(2y-7))(y-5)(x-8)^2+(y-2)^2-0.1)=0 [-30, 32, 47, -16, 24, 16. 25]}

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Answer 2

Avery Alexander

#x^2-16x+6y+43=0#

#"for any point "(x,y)" on the parabola"#

#"the distance from "(x,y)" to the focus and directrix"# #"are equal"#

#"using the "color(blue)"distance formula"" and equating"#

#rArrsqrt((x-8)^2+(y-2)^2)=|y-5|#

#color(blue)"squaring both sides"#

#(x-8)^2+(y-2)^2=(y-5)^2#

#rArrx^2-16x+64+y^2-4y+4=y^2-10y+25#

#rArrx^2-16x+64cancel(+y^2)-4y+4cancel(-y^2)+10y-25=0#

#rArrx^2-16x+6y+43=0#

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Answer 3

Jace Anderson

The equation of the parabola is (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the distance from the vertex to the focus (and from the vertex to the directrix). Given that the focus is at (8,2) and the directrix is y = 5, the vertex is (8, 3), and p = -1. Thus, the equation of the parabola is (x - 8)^2 = -4(y - 3).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.