Question

5) Elimina tetheta from x=2+3 cos theta y=5+3sin theta​

Answer 1

Answer:

Elimina tetheta from x=2+3 cos theta y=5+3sin theta

Answer 2

Question :-

$\rm \: Eliminate \: \theta \: from \\ \\ \rm \: x = 2 + 3cos\theta \: \\ \\\rm \: y \: = 5 + 3sin\theta$

$\large\underline{\sf{Solution-}}$

Given that,

$\rm \: x = 2 + 3cos\theta$

$\rm \: x - 2 = 3cos\theta$

$\rm\implies \:cos\theta = \dfrac{x - 2}{3} - - - (1)$

Also, given that

$\rm \: y = 5 + 3sin\theta$

$\rm \: y - 5 = 3sin\theta$

$\rm\implies \:sin\theta = \dfrac{y - 5}{3} - - - (2)$

We know,

$\rm \: {sin}^{2}\theta + {cos}^{2}\theta = 1$

On substituting the values from equation (1) and (2), we get

$\rm \: {\bigg(\dfrac{x - 2}{3} \bigg) }^{2} + {\bigg(\dfrac{y - 5}{3} \bigg) }^{2} = 1$

$\rm \: \dfrac{ {(x - 2)}^{2} }{9} + \dfrac{ {(y - 5)}^{2} }{9} = 1$

$\rm \: {(x - 2)}^{2} + {(y - 5)}^{2} = 9$

$\rm \: {x}^{2} + 4 - 4x + {y}^{2} + 25 - 10y = 9$

$\rm \: {x}^{2} + {y}^{2} - 4x - 10y + 20 = 0$

$\rule{190pt}{2pt}$

Additional Information :-

$\rm \: {sec}^{2}\theta - {tan}^{2}\theta = 1$

$\rm \: {cosec}^{2}\theta - {cot}^{2}\theta = 1$

$\rule{190pt}{2pt}$

More Identities to know :

➢  (a + b)² = a² + 2ab + b²

➢  (a - b)² = a² - 2ab + b²

➢  a² - b² = (a + b)(a - b)

➢  (a + b)² = (a - b)² + 4ab

➢  (a - b)² = (a + b)² - 4ab

➢  (a + b)² + (a - b)² = 2(a² + b²)

➢  (a + b)³ = a³ + b³ + 3ab(a + b)

➢  (a - b)³ = a³ - b³ - 3ab(a - b)