The equation of parabola with a focus of (8,2) and a directrix of x = 6 is [tex]\bold{(y-2)^2=4(x-7)}[/tex]
"It is a plane curve generated by moving a point so that its distance from a fixed point is equal to its distance from a fixed line."
"The fixed point is the focus of parabola"
"The fixed-line is the directrix of the parabola."
"The distance between two points [tex](x_1,y_1),(x_2,y_2)[/tex] is,
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]"
For given question,
We know, parabola is a curve whose distance from a fixed point is equal to its distance from a fixed line.
We have been given a focus of (8, 2) and a directrix of x = 6
Using distance formula the distance from a fixed point would be,
[tex]d=\sqrt{(x-8)^2+(y-2)^2}[/tex]
And the distance from a fixed line would be,
d1 = x - 6
distance from a fixed point = distance from a fixed line
⇒ [tex]\sqrt{(x-8)^2+(y-2)^2}=x-6[/tex]
⇒ [tex](x-8)^2+(y-2)^2=(x-6)^2[/tex] ..........(squaring on both the sides)
⇒ [tex]x^{2} -16x+64+(y-2)^2=x^{2} -12x+36[/tex]
⇒ [tex](y-2)^2=x^{2} -12x+36-x^{2} +16x-64[/tex]
⇒ [tex](y-2)^2=4x-28[/tex]
⇒ [tex]\bold{(y-2)^2=4(x-7)}[/tex]
Therefore, the equation of parabola with a focus of (8,2) and a directrix of x = 6 is [tex]\bold{(y-2)^2=4(x-7)}[/tex]
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