Q. 165.0( 3 Votes )

The points A (1,

Answer :

The figure of the above condition is as shown below



We know in a parallelogram, the diagonals bisect each other, so O is the midpoint of AC and BD (see the above figure)


Let the coordinates of O be x and y,


As O is midpoint of line BD, so the coordinates of O can be written as



Here x1 and y1 are the coordinates of point B(2,3)


And x2 and y2 are the coordinates of point D( – 4, – 3)


Substituting the values, we get




So, x = – 1, y = 0


So, the point O is ( – 1, 0)


Now we will consider the next diagonal AC,


Here also O ( – 1,0) is the midpoint


So, applying the midpoint formula



Here x1 and y1 are the coordinates of point A (1, – 2).


And x2 and y2 are the coordinates of point C(k,2).


Substituting the values, we get



1 + k = 2 × ( – 1)


k = – 2 – 1


k = – 3


So, the required value of k is – 3.


OR


Given: points (3k – 1, k – 2), (k,k – 7) and (k – 1, – k – 2) are collinear


To find: the value of ‘k’


Explanation: collinear means all the points lie on same line.


Let the three points be A(3k – 1,k – 2), B(k,k – 7) and C(k – 1, – k – 2)


So, A, B, C will form a straight line and not a triangle


So, the area should be equal to 0


i.e., ar(ΔABC) = 0


now we know the area of any triangle is given by:



This should be equal to zero,


From the given points,


x1 = 3k – 1, y1 = k – 2


x2 = k, y2 = k – 7


x3 = k – 1, y3 = – k – 2


Now substituting these values in area formula, we get



[(3k – 1) ((k – 7) – ( – k – 2)) + k(( – k – 2) – (k – 2) ) + (k – 1)((k – 2) – (k – 7))] = 0


[(3k – 1) (k – 7 + k + 2) + k ( – k – 2 – k + 2) + (k – 1) (k – 2 – k + 7)] = 0


[(3k – 1) (2k – 5) + k( – 2k) + (k – 1) (5)] = 0


(3k(2k – 5) – 1(2k – 5)) – 2k2 + 5k – 5 = 0


6k2 – 15k – 2k + 5 – 2k2 + 5k – 5 = 0


4k2 – 12k = 0


4k(k – 3) = 0


4k = 0 or k – 3 = 0


k = 0 or k = 3


Hence the required value of k is 0 and 3.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses