Question

If the points $(a,1),(1,b)$ and $(a-1,b-1)$ are collinear, $\alpha ,\beta$ are respectively the arithmetic and geometric means of $a$ and $b$, then $4\alpha -{\beta }^2$ is equal to

• A. $-1$
• B. $0$
• C. $3$
• D. $2$

Answer 1

The correct option is C $3$

Since, The given points are collinear

$\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=0$

so,$\left|\begin{array}{ccc}a& 1& 1\\ 1& b& 1\\ a-1& b-1& 1\end{array}\right|=0$
On expanding,we get
$a(b-b+1)-1(1-a+1)+1(b-1-ab+b)=0$
$2(a+b)-ab=3$ -----------1)
Now $\alpha =\frac{a+b}{2}and\beta =\sqrt{ab}$ ---------2)

From 1) and 2)
Now $4\alpha -{\beta }^2=3$
$2(a+b)-ab=3$