Question

# If $$A(5, 3)$$, $$B(11, -5)$$ and $$P(12, y)$$ are the vertices of a right angled triangle, right angled at P, then $$y$$ is ______.

A
2,4
B
2,4
C
2,4
D
2,4

Solution

## The correct option is A $$-2, 4$$If A(5,3) ,B(11,-55) and P(12,y) are the vertices of a right angled triangle,then$$AB^{ 2 }=\sqrt { (11-5)^{ 2 }+(-5-3)^{ 2 } } =10\\ BC^{ 2 }=\sqrt { (11-12)^{ 2 }+(-5-y)^{ 2 } } =\sqrt { 1+(-5-y)^{ 2 } } \\ AB^{ 2 }=100\quad BC^{ 2 }=1+(-5-y)^{ 2 }\\ AC^{ 2 }=(5-12)^{ 2 }+(3-y)^{ 2 }=49+(3-y)^{ 2 }\\ AB^{ 2 }=AC^{ 2 }+BC^{ 2 }\\ 100=1+(+5+y)^{ 2 }+49+(3-y)^{ 2 }\\ \Longrightarrow 100=1+25+10y+{ y }^{ 2 }+49+3-6y+{ y }^{ 2 }\\ \Longrightarrow 2{ y }^{ 2 }-4y=16\\ \therefore \quad y=-2,4$$Mathematics

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