[" where "b^(2)=a^(2)(1-e^(2))" and "e<1],[" Find the area of the region in the first quadrant enclosed by the "x" -axis,the line "x=sqrt(3)y" and "],[" the circle "x^(2)+y^(2)=4]

Find the area bounded by the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 and the ordinates x" "=" "0 andx" "=" "a e , where, b^2=a^2(1-e^2) ande" "<" "1 .

Find the area of the region bounded by the ellipse x^2/a^2 +y^2/b^2=1, the ordinates x=0and x= ae where b^2=a^2 (1-e^2),e lt 1

A point moves so that the sum of its distances from (a e ,0)a n d(-a e ,0) is 2a , prove that the equation to its locus is (x^2)/(a^2)+(y^2)/(b^2)=1 , where b^2=a^2(1-e^2)dot

A point moves so that the sum of its distances from (a e ,0)a n d(-a e ,0) is 2a , prove that the equation to its locus is (x^2)/(a^2)+(y^2)/(b^2)=1 , where b^2=a^2(1-e^2)dot

A point moves so that the sum of its distances from (a e ,0)a n d(-a e ,0) is 2a , prove that the equation to its locus is (x^2)/(a^2)+(y^2)/(b^2)=1 , where b^2=a^2(1-e^2)dot

A point moves so that the sum of its distances from (a e ,0)a n d(-a e ,0) is 2a , prove that the equation to its locus is (x^2)/(a^2)+(y^2)/(b^2)=1 , where b^2=a^2(1-e^2)dot

If a and b are two vectors such that 2a+b=e_(1)" and "a+2b=e_(2) , where e_(1)=(1, 1, 1)" and "e_(2)=(1,1,-1) , then the angle between a and b is

If a variable line has its intercepts on the coordinate axes e and e', where (e)/(2) and (e')/(2) are the eccentricities of a hyperbola and its conjugate hyperbola,then the line always touches the circle x^(2)+y^(2)=r^(2), where r=1 (b) 2 (c) 3 (d) cannot be decided

If e_(1) is the eccentricity of the ellipse (x^(2))/(16)+(y^(2))/(b^(2))=1 and e_(2) is the eccentricity of the hyperbola (x^(2))/(9)-(y^(2))/(b^(2))=1 and e_(1)e_(2)=1 then b^(2) is equal to

If alpha and beta be the eccentric angles of the extremities of a focal chord of the hyperbola b^(2)x^(2) - a^(2)y^(2) = a^(2)b^(2) , show that, tan(alpha)/(2)tan(beta)/(2) = -(e-1)/(e+1) , (e-1)/(e+1) (e is the eccentricity of the hyperbola.).