Question 37 (i) [Case Based] - CBSE Class 12 Sample Paper for 2023 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Last updated at May 29, 2023 by Teachoo

In an elliptical sport field the authority wants to design a rectangular soccer field with the maximum possible area. The sport field is given by the graph of x
^{
2
}
/a
^{
2
}
+y
^{
2
}
/b
^{
2
}
=1

Question 37 (
i
)

If the length and the breadth of the rectangular field be 2x and 2y respectively, then find the area function in terms of x.

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Question 37 [Case Based] In an elliptical sport field the authority wants to design a rectangular soccer field with the maximum possible area. The sport field is given by the graph of 𝑥^2/𝑎^2 +𝑦^2/𝑏^2 =1Based on the above information answer the following questions.
Question 37 (i) If the length and the breadth of the rectangular field be 2x and 2y respectively, then find the area function in terms of x.Given equation of ellipse is
𝑥^2/𝑎^2 +𝑦^2/𝑏^2 =1
where Major axis of ellipse is along x-axis
Let ABCD be the rectangle
Where
Length of rectangle = 2x
Breadth of rectangle = 2yBy Symmetry
CP = BP = 1/2 BC
CP = BP = 1/2 × 2𝑦=𝑦
Thus,
y-coordinate of point C = 𝒚
Similarly,
By Symmetry
OP = 1/2 AB
OP = 1/2 × 2𝑥=𝑥
Thus,
x-coordinate of point C = 𝒙
Thus, coordinates of point C = (x, y)
Since point C lies in ellipse, it will satisfy it’s equation
𝑥^2/𝑎^2 +𝑦^2/𝑏^2 =1
𝑦^2/𝑏^2 =1−" " 𝑥^2/𝑎^2
𝑦^2/𝑏^2 =" " (𝑎^2 − 𝑥^2)/𝑎^2
𝑦^2=" " 𝑏^2/𝑎^2 (𝑎^2 − 𝑥^2 )
Taking square root by sides
𝑦=" " √(𝑏^2/𝑎^2 (𝑎^2 − 𝑥^2 ) )
𝒚=(" " 𝒃)/𝒂 √((𝒂^𝟐 − 𝒙^𝟐 ) )
Now,
Area of Rectangle = Length × Breadth
= 2x × 2y
= 4xy
Putting 𝑦=(" " 𝑏)/𝑎 √((𝑎^2 − 𝑥^2 ) )
= 4x × (" " 𝑏)/𝑎 √((𝑎^2 − 𝑥^2 ) )
= (" " 𝟒𝒃)/𝒂 𝒙√((𝒂^𝟐 − 𝒙^𝟐 ) )

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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