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Home » Find the equation for the ellipse that satisfies the given conditions: Centre at (0, 0), major axis on the y-axis and passes through the points (3, 2) and (1, 6).
Find the equation for the ellipse that satisfies the given conditions: Centre at (0, 0), major axis on the y-axis and passes through the points (3, 2) and (1, 6).
CBSE | Class 11 | Conic Sections | Exercise 11.3 | Maths | NCERT
Find the equation for the ellipse that satisfies the given conditions: Centre at (0, 0), major axis on the y-axis and passes through the points (3, 2) and (1, 6).

Given:

Focus at

    \[\left( 0,\text{ }0 \right)\]

, significant pivot on the y-hub and goes through the focuses

    \[\left( 3,\text{ }2 \right)\text{ }and\text{ }\left( 1,\text{ }6 \right).\]

Since the middle is at (0,0) and the significant pivot is on they-axis, the condition of the ellipse will be of the structure

    \[{{x}^{2}}/{{b}^{2}}~+\text{ }{{y}^{2}}/{{a}^{2}}~=\text{ }1\]

, where ‘a’ is the semi-significant hub.

Ellipse goes through focuses

    \[\left( 3,\text{ }2 \right)\text{ }and\text{ }\left( 1,\text{ }6 \right).\]

Thus, by putting the qualities

    \[x\text{ }=\text{ }3\text{ }and\text{ }y\text{ }=\text{ }2\]

, we get,

    \[{{3}^{2}}/{{b}^{2}}~+\text{ }{{2}^{2}}/{{a}^{2}}~=\text{ }1\]

    \[9/{{b}^{2}}~+\text{ }4/{{a}^{2}}\ldots .\text{ }\left( 1 \right)\]

Also, by putting the qualities

    \[x\text{ }=\text{ }1\text{ }and\text{ }y\text{ }=\text{ }6\]

, we get,

    \[{{1}^{1}}/{{b}^{2}}~+\text{ }{{6}^{2}}/{{a}^{2}}~=\text{ }1\]

    \[1/{{b}^{2}}~+\text{ }36/{{a}^{2}}~=\text{ }1\text{ }\ldots .\text{ }\left( 2 \right)\]

On tackling condition

    \[\left( 1 \right)\text{ }and\text{ }\left( 2 \right),\]

we get

    \[{{b}^{2}}~=\text{ }10\text{ }and\text{ }{{a}^{2}}~=\text{ }40.\]

∴ The condition of the ellipse is

    \[~{{x}^{2}}/10\text{ }+\text{ }{{y}^{2}}/40~=\text{ }1\text{ }or\text{ }4{{x}^{2}}~+\text{ }y{{~}^{2}}~=\text{ }40\]

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