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Mathematics

Given A = (x + 2, -2) and B = (11, 6). Find x if AB = 17.

Distance Formula

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Answer

Distance between the given points = (x2−x1)2+(y2−y1)2\sqrt{(x2 - x1)^2 + (y2 - y1)^2}

Distance between A(x + 2, -2) and B(11, 6):

⇒(11−(x+2))2+(6−(−2))2=17⇒(11−x−2)2+(6+2)2=172⇒(9−x)2+82=289⇒81+x2−18x+64=289⇒x2−18x+145=289⇒x2−18x+145−289=0⇒x2−18x−144=0⇒x2−24x+6x−144=0⇒(x2−24x)+(6x−144)=0⇒x(x−24)+6(x−24)=0⇒(x−24)(x+6)=0⇒x=24 and −6⇒ \sqrt{(11 - (x + 2))^2 + (6 - (-2))^2} = 17\\[1em] ⇒ (11 - x - 2)^2 + (6 + 2)^2 = 17^2\\[1em] ⇒ (9 - x)^2 + 8^2 = 289\\[1em] ⇒ 81 + x^2 - 18x + 64 = 289\\[1em] ⇒ x^2 - 18x + 145 = 289\\[1em] ⇒ x^2 - 18x + 145 - 289 = 0\\[1em] ⇒ x^2 - 18x - 144 = 0\\[1em] ⇒ x^2 - 24x + 6x - 144 = 0\\[1em] ⇒ (x^2 - 24x) + (6x - 144) = 0\\[1em] ⇒ x(x - 24) + 6(x - 24) = 0\\[1em] ⇒ (x - 24)(x + 6) = 0\\[1em] ⇒ x = 24 \text{ and } -6

Hence, the values of x are 24 and -6.

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