In a △ A B C , A = 30 ∘ , B = 60 and c=10. Find a and b.

Answer

A, C

Explanation

we have ,
equation of circle `x^(2)+y^(2)=(1)/(2)`
and Equation of Parabola `y^(2)=4`

let the equation of common tangent of porabola and circle is
`y=mx+(1)/(m)`
since , radius of circle `=(1)/(sqrt(2))`
`therefore (1) /(sqrt(2))=|(0+0+(1)/(m))/(sqrt(1+m^(2)))|`
`impliesm^(4)+m^(2)-2=0implies m=+-1`
`therefore `Equation of common tangent are
`y=x+1and y=-x=1`
intesection point of common tangent at `q(-1,0) `
`therefore ` Equation of ellipse `(x^(2))/(1)+(y^(2))/(1//2)=1`
where, `a^(2)= 1,b^(2)=1//2`
Now, ecccentricity `(e) =sqrt(1-(b^(2))/(a^(2)))=sqrt(1-(1)/(2))=(1)/(sqrt(2))`
and length of latusrectum `=(2b^(2))/(a) =(2((1)/(2)))/(1)=1`

`therefore ` Area of shaded region
`=2 int_(1//sqrt(2))^(1)(1)/(sqrt(2))sqrt(1-x^(2))dx`
`= sqrt(2) [ (x)/(2) sqrt(1-x^(2))+(1)/(2) sin ^(-1)x]_(1//sqrt(2))^(1)`
`= sqrt(2)((pi)/(8)-(1)/(4))=(pi-2)/(4sqrt(2))`