If A(2, -5), B(-2, 5), C(x, 3) and D(1, 1) be four points such that AB and CD are perpendicular to each other, find the value of x.
For two lines to be perpendicular, their product of slope must be equal to -1.
Given points are A(2, -5),B(-2, 5) and C(x, 3),D(1, 1)
slope $=\left(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\right)$
$\Rightarrow$ Slope of line $A B$ is equal to
$\left(\frac{5+5}{-2-2}\right)$
$=\left(\frac{10}{-4}\right)$
$=\left(\frac{-5}{2}\right)$
$=-2.5$
And the slope of line CD is equal to
$\left(\frac{1-3}{1-x}\right)$
$=\left(\frac{-2}{1-x}\right)$
Their product must be equal to -1
the slope of line AB×Slope of line CD = -1
$\Rightarrow-2.5 \times\left(\frac{-2}{1-x}\right)=-1 \Rightarrow 5=x-1$
⇒x = 6
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