Question

if the sum of quadratic equation ax2+bx+c=0 is equal to sum of squares of their reciprocals. show that bc2,ca2,ab2 are in a.p

Answer 1

Let P and Q are roots of eqn

P + Q = -b/a

PQ = c/a

a/c to question,

P + Q = 1/P² + 1/Q²

-b/a ={ ( P + Q)²-2PQ }/P²Q²

-b/a = { b²/a² -2c/a)/(c/a)²

-b×c²/a³ = b²/a² -2c/a

2c/a = b²/a² + bc²/a³ = b( ab + c²)/a³

2ca² = b²a + bc²
and as we know that if a,b,c are in ap then
2b(let b be the middle term)= a+c