If b 1 b 2-2 c 1- c 2 and b 1- b 2- c 1- c 2 are all real numbers- then at least one of the equations x 2- b 1 x - c 1-0 and x 2- b 2 x - c 2-0 hasA- real rootsB- purely imaginary rootsC- roots of the form a- ib a- b - R- a b-0D- rational roots

The correct option is A real rootsb_1 b_2 = 2(c_1 + c_2) ...(1) x^2 + b_1x + c_1 = 0 nbsp; D_1=b_1^2 4c_1 nbsp; x^2 + b_2x + c_2 = 0 D_2=b_2^2 4c_2 n ...

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Byju's Answer

Standard XII

Mathematics

Common Roots

If b 1 b 2=2 ...

Question

If $b_{1} b_{2} = 2 (c_{1} + c_{2})$ and $b_{1}, b_{2}, c_{1}, c_{2}$ are all real numbers, then at least one of the equations $x^{2} + b_{1} x + c_{1} = 0$ and $x^{2} + b_{2} x + c_{2} = 0$ has

real roots

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purely imaginary roots

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roots of the form

a + i b (a, b \in R, a b \neq 0)

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rational roots

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Solution

The correct option is A real roots
$b_{1} b_{2} = 2 (c_{1} + c_{2}) . . . (1)$

$x^{2} + b_{1} x + c_{1} = 0$
$D_{1} = b_{1}^{2} - 4 c_{1}$

$x^{2} + b_{2} x + c_{2} = 0$
$D_{2} = b_{2}^{2} - 4 c_{2}$

$D_{1} + D_{2} = b_{1}^{2} + b_{2}^{2} - 4 (c_{1} + c_{2}) = (b_{1} - b_{2})^{2} \geq 0$

Atleast one of them $D_{1}, D_{2}$ is greater than or equal to zero.

Hence one of the quadratic must have real roots.

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Similar questions

Q. If

x^{2} + b_{1} x + c_{1} = 0

and

x^{2} + b_{2} x + c_{2} = 0

are the two quadratic equations such that

b_{1} b_{2} = 2 (c_{1} + c_{2})

and

b_{1}, b_{2}, c_{1}, c_{2} \in R,

then which of the following is always true ?

Q. If

b_{1} b_{2} = 2 (c_{1} + c_{2})

and

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x^{2} + b_{1} x + c_{1} = 0

and

x^{2} + b_{2} x + c_{2} = 0

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x^{2} + b_{1} x + c_{1} = 0

and

x^{2} + b_{2} x + c_{2} = 0

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Q. If the ratio of roots of

a_{1} x^{2} + b_{1} x + c_{1} = 0

be equal to the ratio of the roots of

a_{2} x^{2} + b_{2} x + c_{2} = 0,

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and

a_{3} x^{2} + b_{3} x + c_{3} = 0

has a common root, then show that
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\frac{c_{1} a_{2} + c_{2} a_{1}}{c_{1} a_{2} - c_{2} a_{1}} + \frac{a_{1} a_{2} b_{3}}{a_{3} (a_{1} b_{2} - a_{2} b_{1})} = 0

(ii)

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If b1b2=2(c1+c2) and b1,b2,c1,c2 are all real numbers, then at least one of the equations x2+b1x+c1=0 and x2+b2x+c2=0 has

The correct option is A real rootsb1b2=2(c1+c2) ...(1) x2+b1x+c1=0 D1=b21−4c1 x2+b2x+c2=0 D2=b22−4c2 D1+D2=b21+b22−4(c1+c2) =(b1−b2)2≥0 Atleast one of them D1,D2 is greater than or equal to zero. Hence one of the quadratic must have real roots.

If $b_{1} b_{2} = 2 (c_{1} + c_{2})$ and $b_{1}, b_{2}, c_{1}, c_{2}$ are all real numbers, then at least one of the equations $x^{2} + b_{1} x + c_{1} = 0$ and $x^{2} + b_{2} x + c_{2} = 0$ has