Question

If both the roots of the equation $x^2-2kx+k^2-4=0$ lie between -3 and 5, then which one of the following is correct?

• A. $-2<k<2$
• B. $-5<k<3$
• C. $-3<k<5$
• D. $-1<k<3$

Answer 1

The correct option is D $-1<k<3$

$x^2-2kx+k^2-4=0$

$\therefore x=\frac{-(-2k)\pm \sqrt{(-2k{)}^2-4(1\left)\right(k^2-4)}}{2}$

$\therefore x=\frac{2k\pm \sqrt{4k^2-4k^2+16}}{2}$

$\therefore x=\frac{2k\pm 4}{2}$

$\therefore x=k\pm 2$

Larger root $=k+2$

Smaller root $=k-2$

Roots lie in the interval $(-3,5)$

So, larger root $<5$

$\therefore k+2<5$

$\therefore k<3$

Also, smaller root $>-3$

$\therefore k-2>-3$

$\therefore k>-1$

$\therefore -1<k<3$

So, the answer is option (D)