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Mathematics

Given A = (3, 1) and B = (0, y - 1). Find y if AB = 5.

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(3, 1) and B(0, y - 1):

⇒(0−3)2+((y−1)−1)2=5⇒(−3)2+(y−1−1)2=52⇒9+(y−2)2=25⇒9+y2+4−4y=25⇒y2−4y+13=25⇒y2−4y−25+13=0⇒y2−4y−12=0⇒y2−6y+2y−12=0⇒(y2−6y)+(2y−12)=0⇒y(y−6)+2(y−6)=0⇒(y−6)(y+2)=0⇒y=6 or −2⇒ \sqrt{(0 - 3)^2 + ((y - 1) - 1)^2} = 5\\[1em] ⇒ (-3)^2 + (y - 1 - 1)^2 = 5^2\\[1em] ⇒ 9 + (y - 2)^2 = 25\\[1em] ⇒ 9 + y^2 + 4 - 4y = 25\\[1em] ⇒ y^2 - 4y + 13 = 25\\[1em] ⇒ y^2 - 4y - 25 + 13 = 0\\[1em] ⇒ y^2 - 4y - 12 = 0\\[1em] ⇒ y^2 - 6y + 2y - 12 = 0\\[1em] ⇒ (y^2 - 6y) + (2y - 12) = 0\\[1em] ⇒ y(y - 6) + 2(y - 6) = 0\\[1em] ⇒ (y - 6)(y + 2) = 0\\[1em] ⇒ y = 6 \text{ or } -2

Hence, the values of y are 6 and -2.

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