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If the ends of major axis of ellipse are (5,0) and (-5,0). Find the equation of ellipse in the standard form if one of its focus lies on the line 3x - 5y-9 =0.

Answer:

[tex]\qquad\qquad\boxed{ \sf{ \: \bf \: \dfrac{ {x}^{2} }{25} + \dfrac{ {y}^{2} }{16} = 1 \: }}\\ \\ [/tex]

Step-by-step explanation:

Given that, the ends of major axis of ellipse are (5,0) and (-5,0).

So, it means major axis is x - axis.

Center of ellipse is midpoint of end point of major axis.

So, using Midpoint Formula,

Center of ellipse = (0, 0).

Also, as ends of major axis of ellipse are (5,0) and (-5,0).

So, Length of major axis = 10

[tex]\sf\implies 2a = 10 \: \: \sf\implies \: a = 5 \\ \\ [/tex]

Now, as major axis of ellipse is x - axis. So, focus also lies on x - axis.

As it is given that, focus lies on the line 3x - 5y - 9 =0.

So, for x - axis, y = 0

[tex]\sf \: 3x = 9 \: \sf\implies \: x = 3 \\ \\ [/tex]

[tex]\sf\implies Coordinates \: of \: focus \: = (3,0) \\ \\ [/tex]

[tex]\sf\implies \: ae = 3 \\ \\ [/tex]

We know,

[tex]\sf \: {b}^{2} = {a}^{2}(1 - {e}^{2}) \\ \\ [/tex]

[tex]\sf \: {b}^{2} = {a}^{2} - {a}^{2} {e}^{2} \\ \\ [/tex]

[tex]\sf \: {b}^{2} = {a}^{2} - {(ae)}^{2} \\ \\ [/tex]

[tex]\sf \: {b}^{2} = {5}^{2} - {3}^{2} \\ \\ [/tex]

[tex]\sf \: {b}^{2} = 25 - 9 \\ \\ [/tex]

[tex]\sf\implies \sf \: {b}^{2} = 16 \\ \\ [/tex]

So, Equation of ellipse having center (0, 0) is

[tex]\sf \: \dfrac{ {x}^{2} }{ {a}^{2} } + \dfrac{ {y}^{2} }{ {b}^{2} } = 1 \\ \\ [/tex]

[tex]\sf\implies \bf \: \dfrac{ {x}^{2} }{25} + \dfrac{ {y}^{2} }{16} = 1 \\ \\ [/tex]

[tex]\rule{190pt}{2pt}[/tex]

Additional Information

[tex]\begin{gathered}\boxed{\begin{array}{c|c} \bf & \bf \dfrac{ {x}^{2} }{ {a}^{2} } + \dfrac{ {y}^{2} }{ {b}^{2} } = 1 \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf Vertex & \sf ( \pm \: a, \: 0) \\ \\ \sf Focus & \sf ( \pm \: ae, \: 0)\\ \\ \sf eccentricity & \sf e = \sqrt{1 - \dfrac{ {b}^{2} }{ {a}^{2} } } \\ \\ \sf Length \: of \: major \: axis & \sf 2a\\ \\ \sf Length \: of \: minor \: axis & \sf 2b\\ \\ \sf Length \: of \: latus \: rectum & \sf \dfrac{ {2b}^{2} }{a} \\ \\ \sf Distance \: between \: focus & \sf 2ae \end{array}} \\ \end{gathered}[/tex]