If three distinct numbers a ,   b ,   c are in G.P. and the equations a x 2 + 2 b x + c = 0 and d x 2 + 2 e x + f = 0 have a common root, then which one of the following statements is correct?

Question

If three distinct numbers a , &#160; b , &#160; c are in G.P. and the equations a x 2 + 2 b x + c = 0 and d x 2 + 2 e x + f = 0 have a common root, then which one of the following statements is correct?

Aicharya · Accepted Answer

&#8757; &#8201; a , &#8201; b , &#8201; c are in G.P. &#8658; b 2 = a c &#8230; ( i ) Equation a x 2 + 2 b x + c = 0 Hence, D = 4 b 2 - 4 a c = 0 &#8757; b 2 = a c &#8658; equation has equal roots &#8658; roots are = - b a , &#8201; - b a ( &#8757; sum of roots &#945; + &#945; = - 2 b a ) &#8757; Both equations have one root in common. &#8756; Common root is - b a . It will satisfy equation. &#8756; d - b a 2 + 2 e - b a + f = 0 &#8658; d b 2 - 2 a e b + a 2 f = 0 &#8658; d a c - 2 a e b + a 2 f = 0 &#8757; b 2 = a c &#8658; d c - 2 a e b + a f = 0 &#8658; d c a c - 2 e b a c + a f a c = 0 (dividing by a c ) &#8658; d a - 2 e b b 2 + f c = - &#8658; d a + f c = 2 e b &#8658; d a , e b , f c are in A.P.

If three distinct numbers a ,   b ,   c are in G.P. and the equations a x 2 + 2 b x + c = 0 and d…

Solution

If three distinct numbers a , &#160; b , &#160; c are in G.P. and the equations a x 2 + 2 b x + c = 0 and d…

Solution

If three distinct numbers a , b , c are in G.P. and the equations a x 2 + 2 b x + c = 0 and d…