A common tangent T to the curves C_1:x^24+y^29=1 and C_2:x^242-y^2143=1 does not pass thro…

Question

A common tangent $T$ to the curves $C_1:\dfrac{x^2}4+\dfrac{y^2}9=1$ and $C_2:\dfrac{x^2}{42}-\dfrac{y^2}{143}=1$ does not pass through the fourth quadrant. If $T$ touches $C_1$ at $(x_1,y_1)$ and $C_2$ at $(x_2,y_2)$, then $|2x_1+x_2|$ is equal to \_\_\_\_\_\_\_\_.

Answer: 20

Accepted Answer

Tangent to $C_1$: $y=mx+\sqrt{4m^2+9}$. Tangent to $C_2$: $y=mx+\sqrt{42m^2-143}$. Equal: $4m^2+9=42m^2-143\Rightarrow m^2=4\Rightarrow m=-2$ (m=2 rejected, as it would put $T$ through the fourth quadrant).

$c=\sqrt{4(4)+9}=5$; points of contact: $x_1=\dfrac85,\ x_2=\dfrac{84}5$.

$$|2x_1+x_2|=\left|\frac{16}5+\frac{84}5\right|=20.$$

Solutions · 1