Question

Determine the equations of common tangents to the hyperbolas $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1.$

• A. $y=+x+\sqrt{a^2-b^2}$
• B. $y=-x-\sqrt{a^2-b^2}$
• C. $y=-x-\sqrt{a^3-b^3}$
• D. $y=-x-\sqrt{a^3+b^3}$

Answer 1

Answer: (A), (B)

The correct options are
A $y=+x+\sqrt{a^2-b^2}$
B $y=-x-\sqrt{a^2-b^2}$

Tangent to $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is
$y=m_{1}x\pm \sqrt{(a^2{m}_1^2-b^2)}$ ...(1)
The other hyperbola is $\frac{x^2}{(-b^2)}-\frac{y^2}{(-a^2)}=1$
Any tangent to it is $y=m_{2}x\pm \sqrt{(-b^2)m_2^2-(-a^2)}$
If (1) and (2) are same, then
$m_1=m_2$ and $a^2{m}_1^2-b^2=-b^2{m}_2^2+a^2$
or $a^2{m}_1^2-b^2=a^2-b^2{m}_1^2$
or $(a^2+b^2)m_1^2=a^2+b^2$
$\therefore m_1^2=1\therefore m_1=\pm 1$
Hence the common tangents are $y=\pm x\pm \sqrt{a^2-b^2}$