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# ${\text{Write an equation of the horizontal parabola with the given vertex and passing through the given point}}{\text{.}} \\ {\text{Vertex at }}\left( {4, - 3} \right){\text{ and passing through }}\left( {10, - 6} \right). \\$  Verified
${\text{As you know that general equation of a horizontal parabola is:}} \\ {\left( {y - k} \right)^2} = 4a\left( {x - h} \right){\text{ }} \\ {\text{As Vertex is given to us as }}\left( {4, - 3} \right){\text{, therefore we have}} \\ {\left( {y - \left( { - 3} \right)} \right)^2} = 4a\left( {x - \left( 4 \right)} \right){\text{ }} \\ \Rightarrow {\left( {y + 3} \right)^2} = 4a\left( {x - 4} \right){\text{ }} - Equation(1){\text{ }} \\ {\text{Now, putting }}\left( {10, - 6} \right){\text{ in Equation(1) we get}} \\ {\left( { - 6 + 3} \right)^2}{\text{ }} = {\text{ }}4a\left( {10 - 4} \right) \\ \Rightarrow {\left( { - 3} \right)^2}{\text{ }} = {\text{ }}4a\left( 6 \right){\text{ }} \\ \Rightarrow 9 = 24a \\ \Rightarrow a = \frac{3}{8}{\text{ }} - Equation(2) \\ {\text{Using Equation(2) in Equation(1)}} \\ \Rightarrow {\left( {y + 3} \right)^2} = 4\left( {\frac{3}{8}} \right)\left( {x - 4} \right) \\ \Rightarrow {\left( {y + 3} \right)^2} = \frac{3}{2}\left( {x - 4} \right){\text{ is the required Equation}} \\ {\text{Note: For these kinds of questions we must remember the general equation of a horizontal }} \\ {\text{parabola and then put vertex's coordinates in it to get the equation in only one variable 'a'}}{\text{. }} \\ {\text{Now, Put the coordinates if the point given to be on the parabola and solve to get value of 'a'}}{\text{. }} \\ {\text{Put the value of 'a' in the equation to get desired equation}}{\text{.}} \\$