Question

find the value of t if the points (t,2t),(2t,6t),(3,8) are collinear​

Answer 1

$\underline{\textbf{Given:}}$

$\textsf{(t,2t),(2t,6t),(3,8) are collinear}$

$\underline{\textbf{To find:}}$

$\textsf{The value of 't'}$

$\underline{\textbf{Solution:}}$

$\textsf{Let the given points be A(t,2t), B(2t,6t), C(3,8) }$

$\textsf{Since the points are collinear,}$

$\textsf{Slope of AB= Slope of BC}$

$\mathsf{\dfrac{6t-2t}{2t-t}=\dfrac{8-6t}{3-2t}}$

$\mathsf{\dfrac{4t}{t}=\dfrac{8-6t}{3-2t}}$

$\mathsf{4(3-2t)=8-6t}$

$\mathsf{12-8t=8-6t}$

$\mathsf{12-8=8t-6t}$

$\mathsf{2t=4}$

$\implies\boxed{\mathsf{t=2}}$

$\underline{\textbf{Find more:}}$

The angle between the line 2x+y+1=0 and x+ky+5=0 is 90° then k=  

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Answer 2

Given,

The coordinates of the points are = (t,2t),(2t,6t),(3,8)

The points are collinear.

To find,

The value of 't'.

Solution,

We can simply solve this mathematical problem by using the following mathematical process.

If, the points are collinear, then the triangle formed by those three points will have an area of zero (which indirectly means they will not form a triangle.)

So, the area of the triangle formed by the given three points = √½ [t × (6t-8) + 2t × (8-2t) + 3 × (2t-6t)]

= √½(6t²-8t+16t-4t²+6t-18t) = √½(2t²-4t) = √(t²-2t) sq. unit

If, the points are collinear,

√(t²-2t) = 0

t²-2t = 0 [squaring both sides]

t² = 2t

t²/t = 2

t = 2

Hence, the value of 't' is 2