Question

If the hyperbola $xy=8$ and ellipse $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ have a common tangent, then

• A. $\left|b\right|\le 4$
• B. $0<\left|b\right|>4$
• C. $2<\left|b\right|>4$
• D. $\left|b\right|\ge 6$

Answer 1

Answer: (A)

$\left|b\right|\le 4$

$\frac{x^2}{16}+\frac{y^2}{b^2}=1$

equation of tangent with slope $m$ to ellipse is $y=mx\pm \sqrt{16m^2+b^2}$

$xy=8⟹x(mx\pm \sqrt{16m^2+b^2})=8$

$⟹m{x}^2\pm \left(\sqrt{16m^2+b^2}\right)x-8=0$

$D=0⟹(\sqrt{16m^2+b^2}{)}^2-4(m\left)\right(-8)=0$

$16m^2+32m+b^2=0$

$D\ge 0⟹(32{)}^2-4(16\left)\right(b^2)\ge 0$

$⟹\mid b\mid \le 4$