Question

If x,y are the roots of ax^2+bx+c not equal to 0. then the value of x^-2+y^-2 is ?

Answer 1

Answer:

b²/c² - 2a/c

Note:

★ The possible values of the variable for which the polynomial becomes zero are called its zeros.

★ To find the zeros of the polynomial p(x) , operate on p(x) = 0 .

★ A quadratic polynomial can have atmost two zeros .

★ If α and ß are the zeros of the quadratic polynomial ax² + bx + c , then ;

• Sum of zeros , (α + ß) = -b/a

• Product of zeros , (αß) = c/a

★ If α and ß are the zeros of any quadratic polynomial , then it is given by ;

x² - (α + ß)x + αß

★ If α and ß are the zeros of the quadratic polynomial ax² + bx + c , then they (α and ß) are also the zeros of the quadratic polynomial k(ax² + bx + c) , k≠0.

Solution:

Here,

The given quadratic polynomial is ;

ax² + bx + c .

Also,

It is given that , X and Y are the zeros of the given quadratic polynomial .

Thus,

Sum of zeros will be ;

X + Y = -b/a

Also,

Product of zeros will be ;

X•Y = c/a

Now,

X^(-2) + Y^(-2)

= 1/X² + 1/Y²

= (Y² + X²) / (X•Y)²

= [ (X + Y)² – 2X•Y ] / (X•Y)²

= (X + Y)² / (X•Y)² - 2X•Y / (X•Y)²

= (X + Y)² / (X•Y)² – 2/(X•Y)

= (-b/a)² / (c/a)² – 2/(c/a)

= (b²/a²) / (c²/a²) – 2×(a/c)

= (b²/a²) × (a²/c²) – 2a/c

= b²/c² - 2a/c

Hence,

The required answer is : b²/c² - 2a/c