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Find the Value of P for Which the Points (−5, 1), (1, P) and (4, −2) Are Collinear. - Mathematics

Sum

Find the value of p for which the points (−5, 1), (1, p) and (4, −2) are collinear.

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Solution

Points are collinear means the area of the triangle formed by the collinear points is 0.
Using the area of a triangle = `[ 1/2 [x_1(y_2 − y_3) + x_2( y_3 − y_1) + x_3(y_1 − y_2)]`

=`1/2[ −5(p − ( −2)) + 1( −2 − 1) + 4( 1− p)]`

= `1/2[ −5( p + 2) + 1( −3 ) + 4(1 − p)]`

= `1/2[ −5p − 10 − 3 + 4 − 4p]`

= `1/2[ − 5p − 9 − 4p ]`
Area of triangle will be zero points being collinear

`1/2[ −5p − 4p − 9 ]`=0

`1/2[ −9p − 9 ] = 0`

9p + 9 = 0

p = − 1
Therefore, the value of p = −1.

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