E1-a-b-c-0- if 1 is a root of ax2-bx-c-0- E2-b2-a2-2ac- if sin- and cos- are the roots of ax2-bx-c-0-

The correct option is A E_1 is true, E_2 is trueIf 1 is a root of ax^2+bx+c=0 ⇒ a(1)^2+b(1)+c=0 ⇒ a+b+c=0 If sinθ , cos ...

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Byju's Answer

Standard XII

Mathematics

Euler's Representation

E1:a+b+c=0, i...

Question

$E_{1} : a + b + c = 0$ , if $1$ is a root of $a x^{2} + b x + c = 0$ .
$E_{2} : b^{2} - a^{2} = 2 a c$ , if $sin θ$ and $cos θ$ are the roots of $a x^{2} + b x + c = 0$ .

E_{1}

is true,

E_{2}

is true

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E_{1}

is true,

E_{2}

is false

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E_{1}

is false,

E_{2}

is true

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E_{1}

is false,

E_{2}

is false

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Solution

The correct option is A $E_{1}$ is true, $E_{2}$ is true
If $1$ is a root of $a x^{2} + b x + c = 0$
$\Rightarrow a (1)^{2} + b (1) + c = 0$
$\Rightarrow a + b + c = 0$ If $sin θ, c o s θ$ are roots of $a x^{2} + b x + c = 0$
$\Rightarrow sin θ + cos θ = \frac{- b}{a}$
$sin θ cos θ = \frac{c}{a}$
${(\frac{- b}{a})}^{2} = {sin}^{2} θ + {cos}^{2} θ + 2 sin θ cos θ$ $\frac{b^{2}}{a^{2}} = 1 + \frac{2 c}{a}$
$\frac{b^{2} - a^{2}}{a} = \frac{2 c}{a}$
$b^{2} - a^{2} = 2 a c$

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Similar questions

Q. If

e_{1} a n d e_{2}

are the eccentricities of a hyperbola and its conjugate respectively, Then

e_{1}^{- 2} + e_{2}^{- 2} = 1

Q. If

E_{1}

and

E_{2}

are two events such that

P (E_{1}) = \frac{1}{4}, P (\frac{E_{2}}{E_{1}}) = \frac{1}{2}

and

P (\frac{E_{1}}{E_{2}}) = \frac{1}{4},

then which of the following is/are correct?

Q. The e.m.f of a Daniell cell at

298 K

E_{1}

Z n | Z n S O_{4} (0.01 M) | | C u S O_{4} (1.0 M) | C u

When the concentration of

Z n S O_{4}

1.0 M

and that of

C u S O_{4}

0.01 M

, the e.m.f changed to

E_{2}

. What is the relationship between

E_{1}

and

E_{2}

E_{1} : \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - 1 = 0, (a > b)

and

E_{2} : \frac{x^{2}}{k^{2}} + \frac{y^{2}}{b^{2}} - 1 = 0, (k < b)

is inscribed in

E_{1} .

E_{1}

and

E_{2}

have same eccentricities and length of minor axis of

E_{2} = p \times L L R of E_{1},

then

p =

E_{1} = \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - 1 = 0, (a > b)

and

E_{2} = \frac{x^{2}}{k^{2}} + \frac{y^{2}}{b^{2}} - 1 = 0, (k < b) E_{2}

is inscribed in

E_{1}

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E_{1}

and

E_{2}

have same eccentricities then length of minor axis of

E_{2} = p (

LLR of

E_{1})

then

p = ?

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E1:a+b+c=0, if 1 is a root of ax2+bx+c=0.E2:b2−a2=2ac, if sinθ and cosθ are the roots of ax2+bx+c=0.

The correct option is A E1 is true, E2 is trueIf 1 is a root of ax2+bx+c=0⇒a(1)2+b(1)+c=0⇒a+b+c=0 If sinθ,cosθ are roots of ax2+bx+c=0⇒sinθ+cosθ=−basinθcosθ=ca(−ba)2=sin2θ+cos2θ+2sinθcosθb2a2=1+2cab2−a2a=2cab2−a2=2ac

$E_{1} : a + b + c = 0$ , if $1$ is a root of $a x^{2} + b x + c = 0$ .
$E_{2} : b^{2} - a^{2} = 2 a c$ , if $sin θ$ and $cos θ$ are the roots of $a x^{2} + b x + c = 0$ .