Exam

NTA Abhyas

Year

2020

Subject

Mathematics

Chapter

Hyperbola

NTA Abhyas 2020 · Mathematics · Hyperbola

Let x ^ 2 + y ^ 2 = 4 r ^ 2 and xy = 1 intersects at A and B in...

Let ${x}^{2} + {y}^{2} = 4{r}^{2} and xy = 1$ intersects at $A and B$ in first quadrant. If $AB = \sqrt{14}$ units, then the value of \left|r\right|$ is

Step-by-step solution

Let A = \left(t, \dfrac{1}{t}\right) \text{ and } B = \left(\dfrac{1}{t}, t\right)\left\{$\begin{array}{ll}∵ Bothcurvesaresymmetric \\ aboutliney = x\end{array}$\right\}
then, ${\left(AB\right)}^{2} = {\left(t - \dfrac{1}{t}\right)}^{2} + {\left(\dfrac{1}{t} - t\right)}^{2} = 14$
$⇒ 2\left({t}^{2} + \dfrac{1}{{t}^{2}}\right) = 14 + 4 ⇒ {t}^{2} + \dfrac{1}{{t}^{2}} = 9$
Now, ${\left(OA\right)}^{2} = {t}^{2} + \dfrac{1}{{t}^{2}} = 9 ⇒ {\left(2r\right)}^{2} = 9$
$⇒ \left|2r\right| = 3$ ⇒ \left|r\right| = 1.5