If real numbers x and y satisfy (x+5)2+(y-12)2=(14)2, then the minimum value of √x2+y2 is

Question

If real numbers x and y satisfy (x+5)2+(y-12)2=(14)2, then the minimum value of √x2+y2 is  (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Let x+5=14 cos θ and y-12=14 sin θ ∴ x2+y2=(14 cos θ-5)2+(14 sin θ+12)2 =196+25+144+28(12 sin θ-5 cos θ) =365+28(12 sin θ-5 cos θ) .∴ √x2+y2| min =√365-28 × 13 =√365-364=1

Q. If real numbers $x$ and $y$ satisfy $(x + 5)^{2} + (y - 12)^{2} = (14)^{2}$ , then the minimum value of $x^{2} + y^{2}$ is _____

Let $x + 5 = 14 cos θ$ and $y - 12 = 14 sin θ$
$∴ x^{2} + y^{2} = (14 cos θ - 5)^{2} + (14 sin θ + 12)^{2}$
$= 196 + 25 + 144 + 28 (12 sin θ - 5 cos θ)$
$= 365 + 28 (12 sin θ - 5 cos θ)$
$∴ x^{2} + y^{2} ∣ ∣_{m i n} = 365 - 28 \times 13$
$= 365 - 364 = 1$

Q. If real numbers x and y satisfy (x+5)2+(y−12)2=(14)2, then the minimum value of x2+y2​ is _____

Let x+5=14cosθ and y−12=14sinθ ∴x2+y2=(14cosθ−5)2+(14sinθ+12)2 =196+25+144+28(12sinθ−5cosθ) =365+28(12sinθ−5cosθ) ∴x2+y2​∣∣​min​=365−28×13​ =365−364​=1

Q. If real numbers $x$ and $y$ satisfy $(x + 5)^{2} + (y - 12)^{2} = (14)^{2}$ , then the minimum value of $x^{2} + y^{2}$ is _____

Let $x + 5 = 14 cos θ$ and $y - 12 = 14 sin θ$
$∴ x^{2} + y^{2} = (14 cos θ - 5)^{2} + (14 sin θ + 12)^{2}$
$= 196 + 25 + 144 + 28 (12 sin θ - 5 cos θ)$
$= 365 + 28 (12 sin θ - 5 cos θ)$
$∴ x^{2} + y^{2} ∣ ∣_{m i n} = 365 - 28 \times 13$
$= 365 - 364 = 1$