Question

If (3, 2), (4, k) and (5, 3) are collinear, then k is equal to:
(a) \[\dfrac{3}{2}\]
(b) \[\dfrac{2}{5}\]
(c) \[\dfrac{5}{2}\]
(d) \[\dfrac{3}{5}\]

Answer 1

Hint: First of all look at the definition of collinear. Now, apply the condition to these 3 points to get a relation in k. From this relation, try to get all the constants on the right-hand side. Now, find the coefficient of k. Divide the equation on both sides with this coefficient to get the value of k which is our required result.

Complete step-by-step answer:
Collinear: A group of points is said to be collinear if all the points lie on the same line. From this definition, we can say that all points lie on a line, passing through any two points of those.
For any two points, there is only one line possible to satisfy the condition, both of them lying on this.
The given points which are collinear, are written in the form: (3, 2), (4, k), (5, 3).
By definition, we can say that the (4,k) point lies in the equation of the line passing through (3, 2) and (5, 3).
The equation of the line passing through (a, b) and (c, d) is given by:
\[y-b=\dfrac{d-b}{c-a}\left( x-a \right)\]
Here, we have a = 3, b = 2, c = 5, d = 3.
By substituting these values, we get the equation of line as
\[y-2=\dfrac{3-2}{5-3}\left( x-3 \right)\]
By simplifying the fraction, we can write the equation as
\[y-2=\dfrac{1}{2}\left( x-3 \right)\]
By multiplying with 2 on both the sides, we get it as:
\[2\left( y-2 \right)=\left( x-3 \right)\]
By removing the bracket, we can write it in the form of:
\[2y-4=x-3\]
By adding 4 on both the sides of the equation, we get,
\[2y=x-3+4\]
By simplifying the above equation, we can write it as,
\[2y=x+1\]
By definition, we have (4, k) lies on the equation of the line. By substituting (4, k) into the equation, we can write it as,
\[2\left( k \right)=\left( 4 \right)+1\]
By removing the bracket and simplifying, we get the equation in the form:
\[2k=5\]
By dividing with 2 on both the sides, we get the value of k as
\[\dfrac{2k}{2}=\dfrac{5}{2}\]
By simplifying, we get the value of k as
\[k=\dfrac{5}{2}\]
Hence, the option (c) is the right answer.

Note: While finding the equation, if you do the second point – first point in the numerator, you must do the same on the denominator or else you will get the wrong answer. While substituting the point also, take care that k must be substituted in y and not x. We left the equation of line as y = some form because we know that we need the value of k and if it is in the y – coordinate, if you want x, then you must write the equation as x = f(y) to solve it easily.