[" If "A_(1)B_(1)" and "A_(2)B_(2)" are two focal chords of the parabola "],[y^(2)=4ax," then the chords "A_(1)A_(2)" and "B_(1)B_(2)" intersect on "]

Delta=|[a_(1),b_(1)],[a_(2),b_(2)]| and A_(1),B_(1),A_(2),B_(2) are cofactor of a_(1),b_(1),a_(2),b_(2) respectively then , |[A_(1),B_(1)],[A_(2),B_(2)]| has value equal to

If the points (a_(1),b_(1))m(a_(2),b_(2))" and " (a_(1)-a_(2),b_(2)-b_(2)) are collinear, then prove that a_(1)/a_(2)=b_(1)/b_(2)

If the points (a_(1),b_(1))m(a_(2),b_(2))" and " (a_(1)-a_(2),b_(2)-b_(2)) are collinear, then prove that a_(1)/a_(2)=b_(1)/b_(2)

If A_(1) and A_(2) are two A.M.'s between a and b, prove that (i) (2A_(1)-A_(2))(2A_(2)-A_(1))=ab (ii) A_(1)+A_(2)=a+b

Show that two lines a_(1) x + b_(14) y + c_(1) = 0 and a_(2) x + b_(2) y + c_(2) = 0 , where b_(1) , b_(2) ne 0 are Perpendicular if a_(1) a_(2) + b_(1) b_(2) =0 .

A_(1), A_(2), A_(3), ……, A_(n) " and " B_(1), B_(2), B_(3), …., B_(n) are non-singular square matrices of order n such that A_(1)B_(1) = I_(n), A_(2)B_(2) = I_(n), A_(3)B_(3) = I_(n),……A_(n)B_(n) = I_(n) " then"(A_(1) A_(2)A_(3)….. A_(n))^(-1) = ______.

Let a_(1), a_(2),…. and b_(1),b_(2),…. be arithemetic progression such that a_(1)=25 , b_(1)=75 and a_(100)+b_(100)=100 , then the sum of first hundred term of the progression a_(1)+b_(1) , a_(2)+b_(2) ,…. is equal to