Exam

NTA Abhyas

Year

2020

Subject

Mathematics

Chapter

Ellipse

NTA Abhyas 2020 · Mathematics · Ellipse

If common tangents of x ^ 2 + y ^ 2 = r ^ 2 and x ^ 2 16 + y ^ 2 9...

If common tangents of ${x}^{2} + {y}^{2} = {r}^{2}$ and $\dfrac{{x}^{2}}{16} + \dfrac{{y}^{2}}{9} = 1$ forms a square, then the length of diagonal of the square is

Step-by-step solution

Director circle of ${x}^{2} + {y}^{2} = {r}^{2}$ and $\dfrac{{x}^{2}}{16} + \dfrac{{y}^{2}}{9} = 1$ is same
$⇒$ ${x}^{2} + {y}^{2} = 2{r}^{2} \text{ and } {x}^{2} + {y}^{2} = 16 + 9$ represents same circle
$⇒$ $2{r}^{2} = 25 ⇒ r = \dfrac{5}{\sqrt{2}}$
side of square = $2r = 5\sqrt{2}$
diagonal of square = $\sqrt{2}$(side of square)$= \sqrt{2}\left(5\sqrt{2}\right) = 10$