Answer
Verified
430.2k+ views
Hint: By the given equation and condition we find the roots of the given polynomial. After finding the roots of the polynomial we solve this and comparing with the given polynomial, we obtain the \[\alpha \],\[\beta \],\[\gamma \] values.
Complete step-by-step answer:
We know that the eccentricity of a parabola is \[1\].
Also, the eccentricity of a rectangular hyperbola can be determined and it’s shown below:
Rectangular hyperbola is a hyperbola for which the asymptotes are perpendicular, also called an equilateral hyperbola or right hyperbola. This occurs when the si-major and semi- major are equal.
So \[a = b\]
We know that the eccentricity of hyperbola
\[e = \dfrac{{{{({a^2} + {b^2})}^{\dfrac{1}{2}}}}}{a}\]
In rectangular hyperbola we have \[a = b\]
\[ \Rightarrow e = \dfrac{{{{(2{a^2})}^{\dfrac{1}{2}}}}}{a}\]
\[ \Rightarrow e = \dfrac{{\sqrt 2 a}}{a}\]
Hence, \[e = \sqrt 2 \]
Thus the root of polynomial \[f(x)\] is \[1\], \[\sqrt 2 \] and \[ - \sqrt 2 \]
Now \[f(x) = (x - 1)(x - \sqrt 2 )(x + \sqrt 2 )\]
Expanding the first and second brackets,
\[ \Rightarrow f(x) = ({x^2} - x\sqrt 2 - x + \sqrt 2 )(x + \sqrt 2 )\]
Again expanding barkers, we get a polynomial,
\[ \Rightarrow f(x) = {x^3} - {x^2}\sqrt 2 - {x^2} + \sqrt 2 x + {x^2}\sqrt 2 - 2x - \sqrt 2 x + 2\]
Cancelling terms and rearranging,
\[ \Rightarrow f(x) = {x^3} - {x^2} - 2x + 2\]
Comparing with the coefficients given polynomial,
\[f(x) = {x^3} + \alpha {x^2} + \beta x + \gamma \]
We get that \[\alpha = - 1\], \[\beta = - 2\] and \[\gamma = 2\]
Now we need the value of \[\alpha + \beta + \gamma \], and we know the individual values,
Substituting these we get,
\[ \Rightarrow \alpha + \beta + \gamma = - 1 - 2 + 1\]
\[ \Rightarrow \alpha + \beta + \gamma = - 1\]
So, the correct answer is “Option A”.
Note: In conic section, there is a locus of a point in which the distance to the point and the line are in the constant ratio. That ratio is known as eccentricity. It is denoted by\[e\]. We choose another root as \[ - \sqrt 2 \] because if one root is irrational it occurs in a conjugate pair.
Complete step-by-step answer:
We know that the eccentricity of a parabola is \[1\].
Also, the eccentricity of a rectangular hyperbola can be determined and it’s shown below:
Rectangular hyperbola is a hyperbola for which the asymptotes are perpendicular, also called an equilateral hyperbola or right hyperbola. This occurs when the si-major and semi- major are equal.
So \[a = b\]
We know that the eccentricity of hyperbola
\[e = \dfrac{{{{({a^2} + {b^2})}^{\dfrac{1}{2}}}}}{a}\]
In rectangular hyperbola we have \[a = b\]
\[ \Rightarrow e = \dfrac{{{{(2{a^2})}^{\dfrac{1}{2}}}}}{a}\]
\[ \Rightarrow e = \dfrac{{\sqrt 2 a}}{a}\]
Hence, \[e = \sqrt 2 \]
Thus the root of polynomial \[f(x)\] is \[1\], \[\sqrt 2 \] and \[ - \sqrt 2 \]
Now \[f(x) = (x - 1)(x - \sqrt 2 )(x + \sqrt 2 )\]
Expanding the first and second brackets,
\[ \Rightarrow f(x) = ({x^2} - x\sqrt 2 - x + \sqrt 2 )(x + \sqrt 2 )\]
Again expanding barkers, we get a polynomial,
\[ \Rightarrow f(x) = {x^3} - {x^2}\sqrt 2 - {x^2} + \sqrt 2 x + {x^2}\sqrt 2 - 2x - \sqrt 2 x + 2\]
Cancelling terms and rearranging,
\[ \Rightarrow f(x) = {x^3} - {x^2} - 2x + 2\]
Comparing with the coefficients given polynomial,
\[f(x) = {x^3} + \alpha {x^2} + \beta x + \gamma \]
We get that \[\alpha = - 1\], \[\beta = - 2\] and \[\gamma = 2\]
Now we need the value of \[\alpha + \beta + \gamma \], and we know the individual values,
Substituting these we get,
\[ \Rightarrow \alpha + \beta + \gamma = - 1 - 2 + 1\]
\[ \Rightarrow \alpha + \beta + \gamma = - 1\]
So, the correct answer is “Option A”.
Note: In conic section, there is a locus of a point in which the distance to the point and the line are in the constant ratio. That ratio is known as eccentricity. It is denoted by\[e\]. We choose another root as \[ - \sqrt 2 \] because if one root is irrational it occurs in a conjugate pair.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Difference Between Plant Cell and Animal Cell
Give 10 examples for herbs , shrubs , climbers , creepers
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE