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प्रश्न
The area of the region bounded by the ellipse `x^2/25 + y^2/16` = 1 is ______.
पर्याय
20π sq.unit
20π2 sq.units
16π2 sq.units
25π sq.units
उत्तर
The area of the region bounded by the ellipse `x^2/25 + y^2/16` = 1 is 20π sq.unit.
Explanation:
Given equation of ellipse is `x^2/25 + y^2/16` = 1
⇒ `y^2/16 = 1 - x^2/25`
⇒ y2 = `16/25 (25 - x^2)`
∴ y = `4/5 sqrt(25 - x^2)`
∴ Since the ellipse is symmetrical about the axes.
∴ Required area = `4 xx int_0^5 4/5 sqrt(25 - x^2) "d"x`
= `4 xx 4/5 int_0^5 sqrt((5)^2 - x^2) "d"x`
= `6/5 [x/2 sqrt((5)^2 - x^2) + 25/2 sin^-1 x/5]_0^5`
= `16/5 [0 + 25/2 * sin^-1 (5/5) - 0 - 0]`
= `16/5 [25/2 * sin^-1 (1)]`
= `16/5 [25/2 * pi/2]`
= 20π sq.unit
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