focus is (2, 2), directive is x + y = 9 and eccentricity = 2. - Vidyadip

Question

focus is (2, 2), directive is x + y = 9 and eccentricity = 2.

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Answer

Let S (2, 2) be the focus and P(x, y) be a point on the hyperbola.
Draw PM perpendicular from P on the directrix. Then, by definition
sP = ePM
⇒ sP² = e²PM²
^{$\Rightarrow(\text{x}-2)^2+(\text{y}-2)^2=2^2\Big[\frac{\text{x}+\text{y}-9}{\sqrt{1^2+1^2}}\Big]$}^{$\Big[\because\ \text{e}=\frac{4}{3}\Big]$}
$\Rightarrow\ \text{x}^2+4-4\text{x}+\text{y}^2+4-4\text{y}=\frac{4[\text{x+y}-9]^2}{2}$
⇒ x² + y² - 4x - 4y + 8 = 2[x + y - 9]²
⇒ x² + y² - 4x - 4y + 8 = 2[x + y + (-9) + 2 × x × y + 2 × y × (-9) + 2 × (-9) × x]
⇒ x² + y² - 4x - 4y + 8 = 2[x² + y² + 81 + 2xy - 18y + 18x]
⇒ x² + y² - 4x - 4y + 8 = [2x² + 2y² + 162 + 4xy - 36y + 36x]
⇒ 2x² - x² + 2y² - y² + 4xy - 36x + 4x - 36y + 4y + 162 - 8 = 0
⇒ x² + y² - 4xy - 32x - 32y + 154 = 0
This is the required equation of the hyperbola.

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