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Question
Answer the following:
If a, b, c are in G.P. and ax2 + 2bx + c = 0 and px2 + 2qx + r = 0 have common roots then verify that pb2 – 2qba + ra2 = 0
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Solution
a, b, c are in G.P.
∴ b2 = ac
ax2 + 2bx + c = 0 becomes
`"a"x^2 + 2sqrt("ac")x + "c"` = 0
`(sqrt("a")x + sqrt("c"))^2` = 0
∴ x = `(-sqrt("c"))/sqrt("a")`
∴ ax2 + 2bx + c = 0 and px2 + 2qx + r = 0 have a common root, x = `(-sqrt("c"))/sqrt("a")` Satisfying px2 + 2qx + r = 0
∴ `"p"."c"/"a" + 2"q".((-sqrt("c"))/sqrt("a")) + r` = 0
`"pc" - 2"q"sqrt("ac") + "ra"` = 0
`"p"."b"^2/"a" - 2"qb" + "ra"` = 0 ...`[because "b"^2 = "ac", "c" = "b"^2/"a", sqrt("c") = "b"/sqrt("a"), sqrt("ac") = "b"]`
∴ pb2 – 2qba + ra2 = 0
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