If the ellipse `(x^(2))/(a_(1)^(2))+(y^(2))/(b_(1)^(2))=1(a_(1)^(2)gtb_(1)^(2))` has same eccentricity as that of the ellipse `(x^(2))/(a_(2)^(2))+(y^(2))/(b_(2)^(2))=1(a_(2)^(2)gtb_(2)^(2))` prove that `a_(1)b_(2) =a_(2)b_(1)` .

The ellipse (x^(2))/(169) + (y^(2))/(25) = 1 has the same eccentricity as the ellipse (x^(2))/(a^(2)) + (y^(2))/(b^(2)) = 1 . Find the ratio (a)/(b) .

The asymptotes of the hyperbola (x^(2))/(a_(1)^(2))-(y^(2))/(b_(1)^(2))=1 and (x^(2))/(a_(2)^(2))-(y^(2))/(b_(2)^(2))=1 are perpendicular to each other. Then, (a) a_(1)/a_(2)=b_(1)/b_(2) (b) a_(1)a_(2)=b_(1)b_(2) (c) a_(1)a_(2)+b_(1)b_(2)=0 (d) a_(1)-a_(2)=b_(1)-b_(2)

If the lines x=a_(1)y+b_(1),z=c_(1)y+d_(1) and x=a_(2)y+b_(2),z=c_(2)y+d_(2) are perpendicular, prove that, 1+a_(1)a_(2)+c_(1)c_(2)=0 .

Applying vectors , show that (a_(1)b_(1)+a_(2)b_(2)+a_(3)b_(3))^(2)le (a_(1)^(2)+a_(2)^(2)+a_(3)^(2))(b_(1)^(2)+b_(2)^(2)+b_(3)^(2))

If (1+x+x^(2))^(25)=a_(0)+a_(1)x+a_(2)x^(2)+...+a_(50).x^(50) , then a_(0)+a_(2)+a_(4)+....+a_(50) is

Statement - I: Equation of bisectors of the angles between the liens x=0 and y=0 are y=+-x Statement - II : Equation of the bisectors of the angles between the lines a_(1)x+b_(1)y+c_(1)=0anda_(2)x+b_(2)y+c_(2)=0 are (a_(1)x+b_(1)y+c_(1))/(sqrt(a_(1)^(2)+b_(1)^(2)))=+-(a_(2)x+b_(2)y+c_(2))/(sqrt(a_(2)^(2)+b_(2)^(2))) (Provided a_(1)b_(2)nea_(2)b_(1)andc_(1),c_(2)gt0)

If (1+x+x^(2))^(n)=a_(0)+a_(1)x+a_(2)x^(2)+a_(3)x^(3)+.....+a_(2n)x^(2n)" prove that",a_(0)+a_(2)+a_(4)+.......a_(2n)=(1)/(2)(3^(n)+1) .

If a_(1), a_(2), a_(3),…, a_(2k) are in A.P., prove that a_(1)^(2) - a_(2)^(2) + a_(3)^(2) - a_(4)^(2) +…+a_(2k-1)^(2) - a_(2k)^(2) = (k)/(2k-1)(a_(1)^(2) - a_(2k)^(2)) .

If y=1+(a_(1)/(x-a_(1))+(a_(2)x)/((x-a_(1))(x-a_(2)))+(a_(3)x^(2))/((x-a_(1))(x-a_(2))(x-a_(3))) show that, (dy)/(dx)=(y)/(x)[(a_(1))/(a_(1)-x)+(a_(2))/(a_(2)-x)+(a_(3))/(a_(3)-x)]

If a_(1), a_(2),…,a_(n) are in G.P., then show that (1)/(a_(1)^(2) - a_(2)^(2)) + (1)/(a_(2)^(2) - a_(3)^(2))+...+ (1)/(a_(n-1)^(2) - a_(n)^(2)) = (r^(2))/((1-r^(2))^(2))[(1)/(a_(n)^(2))-(1)/(a_(1)^(2))]