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प्रश्न
Calculate the area bounded by the parabola y2 = 4ax and its latus rectum
उत्तर
Given parabola is y2 = 4ax
Its focus is (a, 0)
Equation of latus rectum is x = a
The parabola is symmetrical about the x-axis
Required area = OAB
= `2 xx int_0^"a" y d"x`
= `2 xx int_0^"a" 2sqrt("a")sqrt(x) "d"x`
= ` sqrt("a") int_0^"a" 2(x)^(1/2) "d"x`
= `4sqrt("a") [(x)^(1/2 + 1)/(1/2 + 1)]_0^"a"`
= `4sqrt("a") [(x)^(3/2)/(3/2)]_0^"a"`
= `4 sqrt("a") xx 2/3 [(x)^(3/2)]_0^"a"`
= `8/3 sqrt("a") [("a")^(3/2) - 0]`
= `8/3 sqrt("a")["a"sqrt("a")]`
= `8/3 "a"^2` sq.units
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