If the equations $x^{2} + b^{2} = 1 - 2 b x$ and $x^{2} + a^{2} = 1 - 2 a x$ have one and only one root common, then $| a - b | =$

1

No worries! We‘ve got your back. Try BYJU‘S free classes today!

0

No worries! We‘ve got your back. Try BYJU‘S free classes today!

2

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

None of the above.

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is C $2$
Only one root is common: let $α$ be the common root
$\frac{α^{2}}{b_{1} c_{2} - b_{2} c_{1}} = \frac{α}{a_{2} c_{1} - a_{1} c_{2}} = \frac{1}{a_{1} b_{2} - a_{2} b_{1}}$

First we have to simplify the equations given in the question

$x^{2} + 2 b x + b^{2} - 1 = 0$ and $x^{2} + 2 a x + a^{2} - 1 = 0$

Above Formula is used to find common root between
$a_{1} x^{2} + b_{1} x + c_{1} = 0$
$a_{2} x^{2} + b_{2} x + c_{2} = 0$

Now comparing Given Equations with above Mentioned standard Equation
we get

$a_{1} = 1, b_{1} = 2 b, c_{1} = b^{2} - 1$ and $a_{2} = 1, b_{2} = 2 a, c_{2} = a^{2} - 1$

Now Putting above values in The given $F o r m u l a$ we get

$\frac{α^{2}}{2 b a^{2} - 2 b - 2 a b^{2} + 2 a} = \frac{α}{a^{2} - 1 - b^{2} + 1} = \frac{1}{2 (1 - b)}$

Now $α^{2} = \frac{b a^{2} - b - a b^{2} + a}{1 - b}$ ......................................(1)

and $α = \frac{- (a + b)}{2}$ ..............................................(2)

Putting Value of $α$ in (1)
we get

$\frac{a^{2} + b^{2} + 2 a b}{4}$ = $\frac{b a^{2} - b - a b^{2} + a}{a - b}$

Now Solving Further We Get

$a^{2} + b^{2} + a b = 3 a b + 4$
$\Rightarrow$ $a^{2} + b^{2} - 2 a b = 4$
$\Rightarrow$ $(a - b)^{2} = 4$

$\Rightarrow$ $| a - b | = 2$

Therefore answer is $(C)$

Suggest Corrections

Similar questions

If $x^{2} - 1 = - b^{2} - 2 b x a n d x^{2} - 1 = - a^{2} - 2 a x$ have exactly one root in common then

Given two quadratic equations $x^{2} - 1 = - b^{2} - 2 b x & x^{2} - 1 = - a^{2} - 2 a x$ have exactly one root in common then select the correct statements.

Q. If

α, β

are the roots of

x^{2} - 2 a x + b^{2} = 0

and

γ, δ

are the roots of

x^{2} - 2 b x + a^{2} = 0

, then

Q. Suppose

a, b, c

are real numbers, and each of the equations

x^{2} + 2 a x + b^{2} = 0

and

x^{2} + 2 b x + c^{2} = 0

has two distinct real roots. Then the equation

x^{2} + 2 c x + a^{2} = 0

has

$x^{2} + a^{2} x + b = 0 a n d x^{2} + x + 1 = 0$ have a common roots which of the following is true.

Join BYJU'S Learning Program

If the equations x2+b2=1−2bx and x2+a2=1−2ax have one and only one root common, then |a−b|=

If the equations $x^{2} + b^{2} = 1 - 2 b x$ and $x^{2} + a^{2} = 1 - 2 a x$ have one and only one root common, then $| a - b | =$