Question #180262

Find the area bounded by the parabola y2 = 4ax and its latus rectum.


1
Expert's answer
2021-05-07T12:29:25-0400



Given parabola is y2=4axy^2 = 4ax


 Its focus is (a,0)(a, 0)


 Equation of latus rectum is x=ax = a


The parabola is symmetrical about the x-axis  


Required area = 2[ Area in the first quadrant between the limits x=0 and x=a]2[\text{ Area in the first quadrant between the limits } x = 0 \text{ and } x = a]



=20aydx=20a4axdx=2(2a)0ax12dx=4a[x3232]0a=4a(83a2)=83a2sq.units=2\int_0^aydx\\[9pt]=2\int_0^a\sqrt{4ax}dx\\[9pt]=2(2\sqrt{a})\int_0^ax^{\frac{1}{2}}dx \\[9pt]=4\sqrt{a}[\dfrac{x^{\frac{3}{2}}}{\frac{3}{2}}]|_0^a \\[9pt]=4\sqrt{a}(\dfrac{8}{3}a^2)=\dfrac{8}{3}a^2 sq. units


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