The axis of the parabola $x^{2} + 2 x y + y^{2} - 5 x + 5 y - 5 = 0$ is

x + y = 0

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x + y - 1 = 0

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x - y + 1 = 0

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x - y = \frac{1}{\sqrt{2}}

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Solution

The correct option is A $x + y = 0$
Given parabola is $x^{2} + 2 x y + y^{2} - 5 x + 5 y - 5 = 0$
$⟹ x^{2} + 2 x y + y^{2} = 5 x - 5 y + 5$
$⟹ (x + y)^{2} = 5 x - 5 y + 5 ⟹ (x + y)^{2} = 5 (x - y + 1)$

for a parabola $(A x + C y)^{2} + D x + E y + F = 0$
the axis is $A x + C y + \frac{A D + C E}{2 (A^{2} + C^{2})} = 0$

$⟹$ Axis is $(x + y) + \frac{- 5 + 5}{2 (1 + 1)} = 0$
$∴$ Axis is $x + y = 0$

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\frac{d y}{d x} + \frac{y^{2} + y + 1}{x^{2} + x + 1} = 0

(x + y + 1) = A (1 - x - y - 2 x y)

A

x^{2} + y^{2} + z^{2} + x + y + z - 1 = 0

x^{2} + y^{2} + z^{2} + x + y + z - 5 = 0

The common chord of the circle $x^{2} + y^{2} + 4 x + 1 = 0$ and $x^{2} + y^{2} + 6 x + 2 y + 3 = 0$ is

x + y = 3

x - y = 1

x - y + 5 = 0

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