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{\text{Find the equation of the parabola if the focus is at}}\left( { - 6, - 6} \right){\text{ and the vertex}} \\

{\text{is at }}\left( { - 2,2} \right). \\

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Answer

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{\text{Let Z}}\left( {{x_1},\;{y_1}} \right){\text{ be the coordinates of the point of intersection of the axis and the directrix of the parabola}}{\text{.}} \\

{\text{Then the vertex V}}\left( { - 2,\;2} \right)\;{\text{is the mid point of the line segment joining Z}}\left( {{x_1},\;{y_1}} \right){\text{ and the focus S}}\left( { - 6,\; - 6} \right). \\

\Rightarrow \dfrac{{{x_1} - 6}}{2} = - 2\; \Rightarrow {x_1} = 2 \\

\& \dfrac{{{y_1} - 6}}{2} = 2 \Rightarrow {y_1} = 10 \\

{\text{Thus the directrix meets the axis at Z}}\left( {2,10} \right). \\

{\text{Let }}{{\text{m}}_1}{\text{ be the slope of axis}}{\text{. Then,}} \\

{{\text{m}}_1}{\text{ = }}\left( {{\text{Slope of the line joining the focus S and vertex V}}} \right) = \dfrac{{ - 6 - 2}}{{ - 6 + 2}} = \dfrac{{ - 8}}{{ - 4}} = 2 \\

\Rightarrow {\text{Slope of the directrix which is perpendicular to axis is }} \\

{\text{m = - }}\dfrac{1}{{{{\text{m}}_1}}} = - \dfrac{1}{2} \\

\Rightarrow {\text{equation of directrix which is passing from }}\left( {2,10} \right){\text{ is}} \\

{\text{y - 10 = - }}\frac{1}{2}\left( {x - 2} \right) \\

\Rightarrow 2y + x - 22 = 0 \\

{\text{Let P}}\left( {x,y} \right){\text{ be a point on parabola}}{\text{. Then,}} \\

{\text{Distance of P from the focus = Perpendicular distance of P from the Directrix }}\left( {{\text{Parabola property}}} \right) \\

\Rightarrow \sqrt {{{\left( {x + 6} \right)}^2} + {{\left( {y + 6} \right)}^2}} = \left| {\dfrac{{2y + x - 22}}{{\sqrt {{2^2} + {1^2}} }}} \right| \\

\Rightarrow {\left( {x + 6} \right)^2} + {\left( {y + 6} \right)^2} = \dfrac{{{{\left( {2y + x - 22} \right)}^2}}}{5} \\

\Rightarrow 5{x^2} + 5{y^2} + 60x + 60y + 360 = 4{y^2} + {x^2} + 484 + 4xy - 44x - 88y \\

\Rightarrow 4{x^2} + {y^2} - 4xy + 104x + 148y - 124 = 0 \\

\Rightarrow {\left( {2x - y} \right)^2} + 4\left( {26x + 37y - 31} \right) = 0 \\

{\text{So, this is your required equation of parabola}}{\text{.}} \\

{\text{NOTE: - In this particular type of questions first find the intersection}} \\

{\text{ point of axis and directrix, then find out equation of directrix}} \\

{\text{ then apply parabola property you will get your answer}}{\text{.}} \\

$

{\text{Let Z}}\left( {{x_1},\;{y_1}} \right){\text{ be the coordinates of the point of intersection of the axis and the directrix of the parabola}}{\text{.}} \\

{\text{Then the vertex V}}\left( { - 2,\;2} \right)\;{\text{is the mid point of the line segment joining Z}}\left( {{x_1},\;{y_1}} \right){\text{ and the focus S}}\left( { - 6,\; - 6} \right). \\

\Rightarrow \dfrac{{{x_1} - 6}}{2} = - 2\; \Rightarrow {x_1} = 2 \\

\& \dfrac{{{y_1} - 6}}{2} = 2 \Rightarrow {y_1} = 10 \\

{\text{Thus the directrix meets the axis at Z}}\left( {2,10} \right). \\

{\text{Let }}{{\text{m}}_1}{\text{ be the slope of axis}}{\text{. Then,}} \\

{{\text{m}}_1}{\text{ = }}\left( {{\text{Slope of the line joining the focus S and vertex V}}} \right) = \dfrac{{ - 6 - 2}}{{ - 6 + 2}} = \dfrac{{ - 8}}{{ - 4}} = 2 \\

\Rightarrow {\text{Slope of the directrix which is perpendicular to axis is }} \\

{\text{m = - }}\dfrac{1}{{{{\text{m}}_1}}} = - \dfrac{1}{2} \\

\Rightarrow {\text{equation of directrix which is passing from }}\left( {2,10} \right){\text{ is}} \\

{\text{y - 10 = - }}\frac{1}{2}\left( {x - 2} \right) \\

\Rightarrow 2y + x - 22 = 0 \\

{\text{Let P}}\left( {x,y} \right){\text{ be a point on parabola}}{\text{. Then,}} \\

{\text{Distance of P from the focus = Perpendicular distance of P from the Directrix }}\left( {{\text{Parabola property}}} \right) \\

\Rightarrow \sqrt {{{\left( {x + 6} \right)}^2} + {{\left( {y + 6} \right)}^2}} = \left| {\dfrac{{2y + x - 22}}{{\sqrt {{2^2} + {1^2}} }}} \right| \\

\Rightarrow {\left( {x + 6} \right)^2} + {\left( {y + 6} \right)^2} = \dfrac{{{{\left( {2y + x - 22} \right)}^2}}}{5} \\

\Rightarrow 5{x^2} + 5{y^2} + 60x + 60y + 360 = 4{y^2} + {x^2} + 484 + 4xy - 44x - 88y \\

\Rightarrow 4{x^2} + {y^2} - 4xy + 104x + 148y - 124 = 0 \\

\Rightarrow {\left( {2x - y} \right)^2} + 4\left( {26x + 37y - 31} \right) = 0 \\

{\text{So, this is your required equation of parabola}}{\text{.}} \\

{\text{NOTE: - In this particular type of questions first find the intersection}} \\

{\text{ point of axis and directrix, then find out equation of directrix}} \\

{\text{ then apply parabola property you will get your answer}}{\text{.}} \\

$

Last updated date: 28th Sep 2023

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