A parabola is the locus of a point, which moves so that its distance from a given line called directrix and a given point called focus, is always equal.
Now, the distance between two pints (x_1,y_1)(x1,y1) and (x_2,y_2)(x2,y2) is given by sqrt((x_2-x_1)^2+(y_2-y_1)^2)√(x2−x1)2+(y2−y1)2 and distance of a point (x_1,y_1)(x1,y1) from a line ax+by+c=0ax+by+c=0 is |(ax_1+by_1+c)/sqrt(a^2+b^2)|∣∣∣ax1+by1+c√a2+b2∣∣∣
Coming to parabola with directrix x=103x=103 or x-103=0x−103=0 and focus (108,41)(108,41), let the point equidistant from both be (x,y)(x,y). The distance of (x,y)(x,y) from x-103=0x−103=0 is
|(x-103)/sqrt(1^2+0^2)|=|(x-103)/1|=|x-103|∣∣
∣∣x−103√12+02∣∣
∣∣=∣∣∣x−1031∣∣∣=|x−103|
and its distance from (108,41)(108,41) is
sqrt((108-x)^2+(41-y)^2)√(108−x)2+(41−y)2
and as the two are equal, equation of parabola would be
(108-x)^2+(41-y)^2=(x-103)^2(108−x)2+(41−y)2=(x−103)2
or 108^2+x^2-216x+41^2+y^2-82y=x^2+103^2-206x1082+x2−216x+412+y2−82y=x2+1032−206x
or 11664+x^2-216x+1681+y^2-82y=x^2+10609-206x11664+x2−216x+1681+y2−82y=x2+10609−206x
or y^2-82y-10x+2736=0y2−82y−10x+2736=0
or 10x=y^2-82y+273610x=y2−82y+2736
or 10x=(y-41)^2+105510x=(y−41)2+1055
or in vertex form x=1/10(x-41)^2+211/2x=110(x−41)2+2112
and vertex is (105 1/2,41)(10512,41)
Its graph appears as shown below, along with focus and directrix.
graph{(y^2-82y-10x+2736)((108-x)^2+(41-y)^2-0.6)(x-103)=0 [51.6, 210.4, -13.3, 66.1]}
