Geo5.1.10

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Obtain the equation of the parabola whose focus is (4,5) and vertex is (3,6)


i).Given the focus S=(4,5),A(3,6)\,

Let SA meet the directrix in Z(h,k). A=mid point of SZ.

(3,6)=\left({\frac  {h+4}{2}},{\frac  {k+5}{2}}\right)\,

h=2,k=7\,

The point is Z(2,7). Slope of SZ=1.

Directrix is perpendicular to SZ.

Therefore, slope of the directrix is -1.

Therefore,the equation to the directrix is y-7=1(x-2),x-y+5=0\,

Let P(x_1,y_1) be any point on the parabola and PM be the perpendicular to the directrix.

SP^{{2}}=PM^{{2}}\,

(x_{1}-4)^{2}+(y_{1}-5)^{2}=({\frac  {x-y+5}{{\sqrt  {2}}}})^{2}\,

x_{1}^{{2}}+y_{1}^{{2}}-26x_{1}-10y_{1}+57=0\,

Hence the locus of P is

x^{2}+y^{2}+2xy-26x-10y+57=0\,


Main Page:Geoemetry:The Parabola