If the lines `a_(1)x+b_(1)y+c_(1)=0` and `a_(2)x+b_(2)y+c_(2)=0` cut the coordinates axes in concyclic points, then prove that `|a_(1)a_(2)|=|b_(1)b_(2)|`.

If the lines a_1x+b_1y+c_1=0 and a_2x+b_2y+c_2=0 cut the coordinae axes at concyclic points, then prove that |a_1a_2|=|b_1b_2|

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

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) 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

If ( a_(1))/(a_(2)) cancel( =) ( b_(1))/(b_(2)) where a_(1) x + b_(1) y + c _(1) = 0 and a_(2) x + b_(2) y + c_(2) = 0 then the given pair of linear equation has "______________" solution (s)

If the quadratic polynomials defined on real coefficient P(x)=a_(1)x^(2)+2b_(1)x+c_(1) and Q(x)=a_(2)x^(2)+2b_(2)x+c_(2) take positive values AA x in R , what can we say for the trinomial g(x)=a_(1)a_(2)x^(2)+b_(1)b_(2)x+c_(1)c_(2) ?

If (a_(1))/(a_(2))=(b_(1))/(b_(2))cancel(=)(c_(1))/(c_(2)) where a_(1)x + b_(1) y + c_(1) = 0 and a_(2) x + b_(2) y + c_(2)= 0 then the given pair of linear equation has "____________" solution (s) .

Repeated roots : If equation f(x) = 0, where f(x) is a polyno- mial function, has roots alpha,alpha,beta,… or alpha root is repreated root, then f(x) = 0 is equivalent to (x-alpha)^(2)(x-beta)…=0, from which we can conclude that f(x)=0 or 2(x-alpha)[(x-beta)...]+(x-alpha)^(2)[(x-beta)...]'=0 or (x-alpha) [2 {(x-beta)...}+(x-alpha){(x-beta)...}']=0 has root alpha . Thus, if alpha root occurs twice in the, equation, then it is common in equations f(x) = 0 and f'(x) = 0. Similarly, if alpha root occurs thrice in equation, then it is common in the equations f(x)=0, f'(x)=0, and f'''(x)=0. 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 have a pair of repeated roots common, then |{:(3a_(1),2b_(1),c_(1)),(3a_(2),2b_(2),c_(2)),(a_(2)b_(1)-a_(1)b_(2),c_(1)a_(2)-c_(2)a_(1),d_(1)a_(2)-d_(2)a_(1)):}|=

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))