this will be done by assuming the values of x and y so by assuming x=3 y²=36 so y=6
like this you have to assume 2 more values for x and y
[tex]\large\underline{\sf{Solution-}}[/tex]
Given parabola is
[tex]\rm \: {y}^{2} = 12x \\ \\ [/tex]
On comparing with
[tex]\bf \: {y}^{2} = 4ax \\ \\ [/tex]
We get,
[tex]\rm \: 4a \: = \: 12 \\ \\ [/tex]
[tex]\rm\implies \:\rm \: a \: = \: 3 \\ \\ [/tex]
Now,
[tex]\rm \: Focus \: = \: (a, \: 0) \\ \\ [/tex]
[tex]\bf\implies \: Focus \: = \: (3, \: 0) \\ \\ [/tex]
[tex]\rule{190pt}{2pt} \\ [/tex]
[tex] { \red{ \mathfrak{Additional\:Information}}}[/tex]
[tex]\begin{gathered}\boxed{\begin{array}{c|c} \bf {y}^{2} = 4ax & \bf \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf Vertex & \sf (0,0) \\ \\ \sf Focus & \sf (a,0) \\ \\ \sf Length \: of \: latus \: rectum & \sf 4a\\ \\ \sf Equation \: of \: axis & \sf y = 0\\ \\ \sf Equation \: of \: directrix & \sf y = - a \end{array}} \\ \end{gathered} \\ [/tex]
[tex]\begin{gathered}\boxed{\begin{array}{c|c} \bf {y}^{2} = - 4ax & \bf \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf Vertex & \sf (0,0) \\ \\ \sf Focus & \sf ( - a,0) \\ \\ \sf Length \: of \: latus \: rectum & \sf 4a\\ \\ \sf Equation \: of \: axis & \sf y = 0\\ \\ \sf Equation \: of \: directrix & \sf y = a \end{array}} \\ \end{gathered} \\ [/tex]
[tex]\begin{gathered}\boxed{\begin{array}{c|c} \bf {x}^{2} = 4ay & \bf \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf Vertex & \sf (0,0) \\ \\ \sf Focus & \sf (0,a) \\ \\ \sf Length \: of \: latus \: rectum & \sf 4a\\ \\ \sf Equation \: of \: axis & \sf x = 0\\ \\ \sf Equation \: of \: directrix & \sf x = - a \end{array}} \\ \end{gathered} \\ [/tex]
[tex]\begin{gathered}\boxed{\begin{array}{c|c} \bf {x}^{2} = \: - \: 4ay & \bf \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf Vertex & \sf (0,0) \\ \\ \sf Focus & \sf (0, - a) \\ \\ \sf Length \: of \: latus \: rectum & \sf 4a\\ \\ \sf Equation \: of \: axis & \sf x = 0\\ \\ \sf Equation \: of \: directrix & \sf x = a \end{array}} \\ \end{gathered} \\ [/tex]
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Verified answer
this will be done by assuming the values of x and y so by assuming x=3 y²=36 so y=6
like this you have to assume 2 more values for x and y
[tex]\large\underline{\sf{Solution-}}[/tex]
Given parabola is
[tex]\rm \: {y}^{2} = 12x \\ \\ [/tex]
On comparing with
[tex]\bf \: {y}^{2} = 4ax \\ \\ [/tex]
We get,
[tex]\rm \: 4a \: = \: 12 \\ \\ [/tex]
[tex]\rm\implies \:\rm \: a \: = \: 3 \\ \\ [/tex]
Now,
[tex]\rm \: Focus \: = \: (a, \: 0) \\ \\ [/tex]
[tex]\bf\implies \: Focus \: = \: (3, \: 0) \\ \\ [/tex]
[tex]\rule{190pt}{2pt} \\ [/tex]
[tex] { \red{ \mathfrak{Additional\:Information}}}[/tex]
[tex]\begin{gathered}\boxed{\begin{array}{c|c} \bf {y}^{2} = 4ax & \bf \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf Vertex & \sf (0,0) \\ \\ \sf Focus & \sf (a,0) \\ \\ \sf Length \: of \: latus \: rectum & \sf 4a\\ \\ \sf Equation \: of \: axis & \sf y = 0\\ \\ \sf Equation \: of \: directrix & \sf y = - a \end{array}} \\ \end{gathered} \\ [/tex]
[tex]\begin{gathered}\boxed{\begin{array}{c|c} \bf {y}^{2} = - 4ax & \bf \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf Vertex & \sf (0,0) \\ \\ \sf Focus & \sf ( - a,0) \\ \\ \sf Length \: of \: latus \: rectum & \sf 4a\\ \\ \sf Equation \: of \: axis & \sf y = 0\\ \\ \sf Equation \: of \: directrix & \sf y = a \end{array}} \\ \end{gathered} \\ [/tex]
[tex]\begin{gathered}\boxed{\begin{array}{c|c} \bf {x}^{2} = 4ay & \bf \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf Vertex & \sf (0,0) \\ \\ \sf Focus & \sf (0,a) \\ \\ \sf Length \: of \: latus \: rectum & \sf 4a\\ \\ \sf Equation \: of \: axis & \sf x = 0\\ \\ \sf Equation \: of \: directrix & \sf x = - a \end{array}} \\ \end{gathered} \\ [/tex]
[tex]\begin{gathered}\boxed{\begin{array}{c|c} \bf {x}^{2} = \: - \: 4ay & \bf \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf Vertex & \sf (0,0) \\ \\ \sf Focus & \sf (0, - a) \\ \\ \sf Length \: of \: latus \: rectum & \sf 4a\\ \\ \sf Equation \: of \: axis & \sf x = 0\\ \\ \sf Equation \: of \: directrix & \sf x = a \end{array}} \\ \end{gathered} \\ [/tex]