Question

Consider two straight lines, each of which is tangent to both the circle `x^2+y^2=1/2` and the parabola `y^2=4x` . Let these lines intersect at the point `Q` . Consider the ellipse whose center is at the origin `O(0, 0)` and whose semi-major axis is `O Q` . If the length of the minor axis of this ellipse is `sqrt(2)` , then which of the following statement(s) is (are) TRUE? For the ellipse, the eccentricity is `1/(sqrt(2))` and the length of the latus rectum is 1 (b) For the ellipse, the eccentricity is `1/2` and the length of the latus rectum is `1/2` (c) The area of the region bounded by the ellipse between the lines `x=1/(sqrt(2))` and `x=1` is `1/(4sqrt(2))(pi-2)` (d) The area of the region bounded by the ellipse between the lines `x=1/(sqrt(2))` and `x=1` is `1/(16)(pi-2)`
A. for the ellipse , the eccentricity is `1//sqrt(2)` and the length of the latus rectum is 1.
B. for the ellipse , the eccentricity is `1//2` and the length of the Latus is 1//2
C. the area of the region bounded by the ellipse between the lines `x=(1)/(2) and X=1 " is " (1) /(4sqrt(2))(pi-2)`
D. the area of the region bounded by the ellipse between the lines `x=(1)/(sqrt(2)) and x=1 " is " (1) /(16) (pi-2)`

Answer 1

Correct Answer - A::C
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))`