If the curves `ax^(2) + by^(2) =1 and a_(1) x^(2) + b_(1) y^(2) = 1` intersect each other orthogonally then show that `(1)/(a) - (1)/(b) = (1)/(a_(1)) - (1)/(b_(1))`

The curves 2x^(2) + 3y^(2) = 1 and cx^(2) + 4y^(2) = 1 cut each other orthogonally then the value of c is:

If (1+x+x^(2))^(20) = a_(0) + a_(1)x^(2) "……" + a_(40)x^(40) , then following questions. The value of a_(0)^(2) - a_(1)^(2) + a_(2)^(2)- "……" - a_(19)^(2) is

If a_(1) x^(3) + b_(1)x^(2) + c_(1)x + d_(1) = 0 and a_(2)x^(3) + b_(2)x^(2) + c_(2)x + d_(2) = 0 a pair of repeated roots common, then prove that |{:(3a_(1)", "2b_(1) ", "c_(1)),(3a_(2)", " 2b_(2)", "c_(1)),(a_(2)","b_(1)- a_(1)b_(2)", "c_(2)a_(1)-c_(2)a_(1)", "d_(1)a_(2)-d_(2)a_(1)):}|=0

The line a_(1)x+b_(1)y+c_(1)=0 and a_(2)x+b_(2)y+c_(2)=0 are perpendicular if:

Two A.M's A_(1) and A_(2) , two G.M's G_(1) and G_(2) and two H.M's H_(1) and H_(2) are inserted between two numbers a and b . (i) Express A_(1) + A_(2) in terms of a and b . (ii) Express G_(1) G_(2) in terms of a and b . (iii) Express 1/H_(1) + 1/H_(2)1 in terms of a and b . (iv) Show that 1/H_(1) + 1/h_(2) = (A_(1) + A_(2))/(G_(1) G_(2))

If (1+x+x^(2))^(20) = a_(0) + a_(1)x^(2) "……" + a_(40)x^(40) , then following questions. The value of a_(0) + a_(1) + a_(2) + "……" + a_(19) is

Show that two lines a_(1)x + b_(1) y+ c_(1) = 0 " and " a_(2)x + b_(2) y + c_(2) = 0 " where " b_(1) , b_(2) ne 0 are : (i) Parallel if a_(1)/b_(1) = a_(2)/b_(2) , and (ii) Perpendicular if a_(1) a_(2) + b_(1) b_(2) = 0 .

If a_(1), a_(2) , ……. A_(n) are in H.P., then the expression a_(1)a_(2) + a_(2)a_(3) + ….. + a_(n - 1)a_(n) is equal to

If (1+x+x^(2))^(20) = a_(0) + a_(1)x^(2) "……" + a_(40)x^(40) , then following questions. The value of a_(0) + 3a_(1) + 5a_(2) + "……" + 81a_(40) is

If a_(1)=1, a_(2)=1 and a_(n)=2a_(n-1)+a_(n-2), nge3, ninN , then find the first six terms of the sequence.