Question

$
  {\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). \\
$

Answer 1

$

  {\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{.}} \\
$