The Equation of parabola with focus (3,1) and directrix y = 3 is [tex]10x^2 + 9y^2-60x -14y+91 =0[/tex]
A parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
Here, focus (3,1) and directrix y=3
Let the locus of parabola be (x, y)
We know that, in Parabola
Distance between locus to focus = Distance between locus to directrix line
[tex]\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }[/tex] = [tex]\frac{Ax_{1}+By_{1}+C }{\sqrt{A^2 + B^2 + C^2} }[/tex]
[tex]\sqrt{(x-3)^2+(y-1)^2} = \frac{0.x+1.y-3}{\sqrt{0^2+1^2+(-3)^2} }[/tex]
On squaring both sides, we get
[tex](x-3)^2 + (y-1)^2 = (\frac{y-3}{\sqrt{10} } )^2[/tex]
[tex]x^2 -6x + 9 + y^2 -2y + 1 = \frac{y^2 -6y +9}{10}[/tex]
[tex]10x^2 + 9y^2-60x -14y+91 =0[/tex]
Thus, The Equation of parabola with focus (3,1) and directrix y = 3 is [tex]10x^2 + 9y^2-60x -14y+91 =0[/tex]
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