find-the-range-of-x-y-such-that-x-2-2-y-4-2-49-

Question Number 176501 by infinityaction last updated on 20/Sep/22

find the range of x + y such that

{(x - 2)}^{2} + {(y - 4)}^{2} = 49

Commented by cortano1 last updated on 20/Sep/22

let x + y = k \Rightarrow x + y - k = 0

is tangent to circle

so 7 = \frac{∣ 2 + 4 - k ∣}{\sqrt{2}}

\Rightarrow∣ k - 6 ∣ = 7 \sqrt{2}

\Rightarrow - 7 \sqrt{2} ⩽ k - 6 ⩽ 7 \sqrt{2}

\Rightarrow 6 - 7 \sqrt{2} ⩽ k ⩽ 6 + 7 \sqrt{2}

Therefore range of x + y

is [6 - 7 \sqrt{2}, 6 + 7 \sqrt{2}]

Answered by Peace last updated on 20/Sep/22

{\begin{cases} x = 2 + 7 c o s (t) \\ y = 4 + 7 s i n (t) \end{cases} t \in [0, 2 π [

x + y = 6 + 7 (s i n (t) + c o s (t)) = 6 + 7 \sqrt{2} (s i n (t + \frac{π}{4}))

x + y \in [6 - 7 \sqrt{2}, 6 + 7 \sqrt{2}]

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