First slide

Cartesian plane

Question

Without changing the direction of coordinates axes, origin is transferred to $(α, β)$ so that the linear terms in the equation x2 + y2 + 2x – 4y + 6 = 0 are eliminated. The point $(α, β)$ is

Moderate

A

(–1, 2)
B

(1, –2)
C

(1, 2)
D

(–1, –2)

Solution

The given equation is
x² + y² + 2x – 4y + 6 = 0 (1)
Putting x = x′ + $α$ , y = y′ + $β$ in (1), we get
x′² + y′² + x′(2 $α$ + 2) + y′(2 $β$ – 4) + ( $α$ ² + $β$ ²+ 2 $α$ – 4 $β$ + 6) = 0
To eliminate linear terms, we should have 2 $α$ + 2 = 0 and 2 $β$ – 4 = 0
⇒ $α$ = –1 and $β$ = 2
$∴$ ( $α$ , $β$ ) ≡ (–1, 2)

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