Question

if alpha and beta are zeroes of the polynomial x^2+5x-66 then the quadratic polynomial whose zeroes are 1/alpha and 1/beta are
a) 5x^2 +66x +1
b)5x^2 -66x -1
c)66x^2-5x-1
d)66x^2+12x-1​

Answer 1

p(x) = x² + 5x - 66

Zeroes of p(x) are ɑ and β

sum of zeroes = -(coefficient of x ) / (coefficient of x² )

==> ɑ + β = -5

==> 1/ɑ + 1/β = -1/5 ...(1)

Product of zeroes = (constant term ) / (coefficient of x² )

==> ɑ * β = -66

==> 1/ɑ * 1/β = -1/66 ...(2)

We know,

==> p(x) = x² - ( sum of zeroes )x + (product of zeroes)

==> p(x) = x² - (-1/5)x + (-1/66)

==> p(x) = x² + 1/5x - 1/66

Multiply by 66

==> p(x) = 66x² + 66/5x - 1

Multiply by 5

==> p(x) = 330x² + 66x - 5

Answer 2

Answer:

The correct answer is option(c)

Step-by-step explanation:

Given

α and β are the roots of x² + 5x -66

To find,

The quadratic polynomial whose roots are $\frac{1}{\alpha } , \frac{1}{\beta }$

Recall the concepts

If α and β are the roots of the equation ax² + bx +c = 0, then the sum of roots =α + β = ($\frac{-b}{a}$) and product of roots =αβ=  $\frac{c}{a}$ ----------------(A)

The quadratic equation whose roots are α and β is given by

x²  - (α+β)x +αβ -= 0----------------------------(B)

Solution:

Since  α and β of the polynomial x² + 5x -66 = 0, from equation (A) we get,

the sum of roots = α + β = -5

and product of roots = αβ = -66

The quadratic equation whose roots are  $\frac{1}{\alpha } , \frac{1}{\beta }$ is given by

x²  -( $\frac{1}{\alpha } + \frac{1}{\beta }$)x +$\frac{1}{\alpha } \frac{1}{\beta }$  = 0----------------------(1) (From equations (B))

$\frac{1}{\alpha } + \frac{1}{\beta }$ = $\frac{\alpha +\beta }{\alpha \beta }$ = $\frac{-5}{-66}$ = $\frac{5}{66}$

$\frac{1}{\alpha } \frac{1}{\beta }$  = $\frac{1}{-66}$ = $\frac{-1}{66}$

Then (1) becomes

x²  - $\frac{5}{66}$x + $\frac{-1}{66}$ = 0

66x² - 5x -1 = 0

∴ The quadratic polynomial whose roots are $\frac{1}{\alpha } , \frac{1}{\beta }$ = 66x² - 5x -1

∴ The correct answer is option(c)

#SPJ2