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Question Number 181394 by yaojun2t last updated on 24/Nov/22

	Answered by FelipeLz last updated on 25/Nov/22
	$p: parabola d: directrix r: line passing through F s: line passing through O { ((F = (f, 0))),((O = (0, 0) )) :}→ d: x = −f p: (x−f)^2 +(y−0)^2 = (x−(−f))^2 +(y−y)^2 p: (x−f)^2 +y^2 = (x+f)^2 p: y^2 = 4fx A ∈ p → A = (a, 2(√(af))) B ∈ p → B = (b, −2(√(bf))) r: y = mx+n F ∈ r → 0 = mf+n ⇒ n = −mf A ∈ r → 2(√(af)) = ma−mf ⇒ m = ((2(√(af)))/(a−f)) B ∈ r → −2(√(bf)) = mb−mf ⇒ m = ((2(√(bf)))/(f−b)) m = m ((2(√(af)))/(a−f)) = ((2(√(bf)))/(f−b)) f(√a)−b(√a) = a(√b)−f(√b) f((√a)+(√b)) = a(√b)+b(√a) f((√a)+(√b)) = (√(ab))((√a)+(√b)) f = (√(ab)) s: y = ux+v O ∈ s → 0 = u×0+v ⇒ v = 0 A ∈ s → 2(√(af)) = ua+0 ⇒ u = 2(√(f/a)) = 2((b/a))^(1/4) D = (−f, 2((b/a))^(1/4) ×(−f)) D = (−(√(ab)), −2((b/a))^(1/4) ×(√(ab))) D = (−(√(ab)), −2((b/a))^(1/4) ×((a^2 b^2 ))^(1/4) ) D = (−(√(ab)), −2((ab^3 ))^(1/4) ) B = (b, −2(√(bf))) = (b, −2((ab^3 ))^(1/4) ) y_B = y_D$
	$p : p a r a b o l a$ $d : d i r e c t r i x$ $r : l i n e p a s s i n g t h r o u g h F$ $s : l i n e p a s s i n g t h r o u g h O$ ${\begin{cases} F = (f, 0) \\ O = (0, 0) \end{cases} \to d : x = - f$ $p : {(x - f)}^{2} + {(y - 0)}^{2} = {(x - (- f))}^{2} + {(y - y)}^{2}$ $p : {(x - f)}^{2} + y^{2} = {(x + f)}^{2}$ $p : y^{2} = 4 f x$ $A \in p \to A = (a, 2 \sqrt{a f})$ $B \in p \to B = (b, - 2 \sqrt{b f})$ $r : y = m x + n$ $F \in r \to 0 = m f + n \Rightarrow n = - m f$ $A \in r \to 2 \sqrt{a f} = m a - m f \Rightarrow m = \frac{2 \sqrt{a f}}{a - f}$ $B \in r \to - 2 \sqrt{b f} = m b - m f \Rightarrow m = \frac{2 \sqrt{b f}}{f - b}$ $m = m$ $\frac{2 \sqrt{a f}}{a - f} = \frac{2 \sqrt{b f}}{f - b}$ $f \sqrt{a} - b \sqrt{a} = a \sqrt{b} - f \sqrt{b}$ $f (\sqrt{a} + \sqrt{b}) = a \sqrt{b} + b \sqrt{a}$ $f (\sqrt{a} + \sqrt{b}) = \sqrt{a b} (\sqrt{a} + \sqrt{b})$ $f = \sqrt{a b}$ $s : y = u x + v$ $O \in s \to 0 = u \times 0 + v \Rightarrow v = 0$ $A \in s \to 2 \sqrt{a f} = u a + 0 \Rightarrow u = 2 \sqrt{\frac{f}{a}} = 2 \sqrt[4]{\frac{b}{a}}$ $D = (- f, 2 \sqrt[4]{\frac{b}{a}} \times (- f))$ $D = (- \sqrt{a b}, - 2 \sqrt[4]{\frac{b}{a}} \times \sqrt{a b})$ $D = (- \sqrt{a b}, - 2 \sqrt[4]{\frac{b}{a}} \times \sqrt[4]{a^{2} b^{2}})$ $D = (- \sqrt{a b}, - 2 \sqrt[4]{{a b}^{3}})$ $B = (b, - 2 \sqrt{b f}) = (b, - 2 \sqrt[4]{{a b}^{3}})$ $y_{B} = y_{D}$
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