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Question

If the quadratic equation ax2+bx+c=0(a>0) has sec2θ and cosec2θ as its roots, then which of the following must hold good?

A
b+c=0
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B
b24ac0
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C
c4a
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D
4a+b0
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Solution

The correct options are
A b+c=0
B b24ac0
C c4a
Given, the quadratic equation ax2+bx+c=0(a>0).......... (1)
has sec2θ and cosec2θ as its roots,
a(xsec2θ)(xcosec2 θ)=0
ax2a(sec2θ+cosec2θ)x+asec2θcosec2θ=0
b=a(sec2θ+cosec2θ) and c=asec2θcosec2θ

But sec2θ+cosec2θ=sec2θcosec2θ
Sum of the roots is equal to their product and the roots are real.
Hence,
ba=ca
b+c=0
Also,
b24ac0
c24ac0
c(c4a)0
c4a0(c>0)
c4a
Further
b2+4ab0
b+4a0(b<0)
Options A,B and C are correct.

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