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{\text{Write an equation of the horizontal parabola with the given vertex and passing through the given point}}{\text{.}} \\

{\text{Vertex at }}\left( { - 8,2} \right){\text{ and passing through }}\left( {2, - 3} \right). \\

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Answer

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{\text{As you know that general equation of a horizontal parabola is:}} \\

{\left( {y - k} \right)^2} = 4a\left( {x - h} \right){\text{ }} \\

{\text{As Vertex is given to us as }}\left( { - 8,2} \right){\text{, therefore we have}} \\

{\left( {y - 2} \right)^2} = 4a\left( {x - \left( { - 8} \right)} \right){\text{ }} \\

\Rightarrow {\left( {y - 2} \right)^2} = 4a\left( {x + 8} \right){\text{ }} - Equation(1){\text{ }} \\

{\text{Now, putting }}\left( {2, - 3} \right){\text{ in Equation(1) we get}} \\

{\left( { - 3 - 2} \right)^2}{\text{ }} = {\text{ }}4a\left( {2 + 8} \right) \\

\Rightarrow {\left( { - 5} \right)^2}{\text{ }} = {\text{ }}4a\left( {10} \right){\text{ }} \\

\Rightarrow 25 = 40a \\

\Rightarrow a = \frac{5}{8}{\text{ }} - Equation(2) \\

{\text{Using Equation(2) in Equation(1)}} \\

\Rightarrow {\left( {y - 2} \right)^2} = 4\left( {\frac{5}{8}} \right)\left( {x + 8} \right) \\

\Rightarrow {\left( {y - 2} \right)^2} = \frac{5}{2}\left( {x + 8} \right){\text{ is the required Equation}} \\

{\text{Note: For these kinds of questions we must remember the general equation of a horizontal }} \\

{\text{parabola and then put vertex's coordinates in it to get the equation in only one variable 'a'}}{\text{. }} \\

{\text{Now, Put the coordinates if the point given to be on the parabola and solve to get value of 'a'}}{\text{. }} \\

{\text{Put the value of 'a' in the equation to get desired equation}}{\text{.}} \\

$

{\text{As you know that general equation of a horizontal parabola is:}} \\

{\left( {y - k} \right)^2} = 4a\left( {x - h} \right){\text{ }} \\

{\text{As Vertex is given to us as }}\left( { - 8,2} \right){\text{, therefore we have}} \\

{\left( {y - 2} \right)^2} = 4a\left( {x - \left( { - 8} \right)} \right){\text{ }} \\

\Rightarrow {\left( {y - 2} \right)^2} = 4a\left( {x + 8} \right){\text{ }} - Equation(1){\text{ }} \\

{\text{Now, putting }}\left( {2, - 3} \right){\text{ in Equation(1) we get}} \\

{\left( { - 3 - 2} \right)^2}{\text{ }} = {\text{ }}4a\left( {2 + 8} \right) \\

\Rightarrow {\left( { - 5} \right)^2}{\text{ }} = {\text{ }}4a\left( {10} \right){\text{ }} \\

\Rightarrow 25 = 40a \\

\Rightarrow a = \frac{5}{8}{\text{ }} - Equation(2) \\

{\text{Using Equation(2) in Equation(1)}} \\

\Rightarrow {\left( {y - 2} \right)^2} = 4\left( {\frac{5}{8}} \right)\left( {x + 8} \right) \\

\Rightarrow {\left( {y - 2} \right)^2} = \frac{5}{2}\left( {x + 8} \right){\text{ is the required Equation}} \\

{\text{Note: For these kinds of questions we must remember the general equation of a horizontal }} \\

{\text{parabola and then put vertex's coordinates in it to get the equation in only one variable 'a'}}{\text{. }} \\

{\text{Now, Put the coordinates if the point given to be on the parabola and solve to get value of 'a'}}{\text{. }} \\

{\text{Put the value of 'a' in the equation to get desired equation}}{\text{.}} \\

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Last updated date: 03rd Oct 2023

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