Question

Let $f(x)$ and $g(x)$ be two polynomials of degree 2 defined as $f(x)=x^2-2(k-1) x+1$ and $g ( x )= kax ^2+ bx + c$. Given that $a ^2+ b ^2+ c ^2-2 a +6 b -4 c +14=0, a , b , c \in R$ then
The number of value(s) of $k$ for which $f(x)=0$ and $g(x)=0$ has 3 real and distinct roots provided that their is no common solution among the 2 equation is/are

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (A)

$( a -1)^2+( b +3)^2+( c -2)^2=0$
Hence, $a =1 ; b =-3 ; c =2$
$f ( x )= x ^2-2( k -1) x +1$
$g ( x )= kx ^2-3 x +2$
Let $D _1$ be discriminant of $f ( x )=0$ and $D _2$ be discriminant of $g ( x )=0$
$D_1=4(k-1)^2-4 ; D_2=(9-8 k)$
Now,
Case-I : $D_1>0$ and $D_2=0$
$(k-1)^2-1>0 \text { and } k=\frac{9}{8}$
Hence, no value of $k$ from this case.
Case-II : $D _2>0$ and $D _1=0$
$9-8 k >0$ and $( k -1)^2-1=0$
From this case $k=0$ which is rejected.
Hence, no value of $k$ from this case.
Therefore no value of k is nossible.