Question

for the value of k the point (1,-1),(k,1) and (4,5) are collinear

Answer 1

Step-by-step explanation:

Consider the given points.

(1,−1),(k,1) and (4,5)

Since, these points are collinear means that the area of triangle must me zero.

So,

½∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣=0

where (x1,y1),(x2,y2),(x2,y3) are the points

Therefore,

1(1−5)+k(5+1)+4(−1−1)=0

1(−4)+k(6)+4(−2)=0

−4+12k−8=0

−4+4=0

4k=4

k=1

Hence, this is the answer.

Answer 2

Concept:

One of the most intriguing ideas in mathematics is called coordinate geometry. By using graphs with curves and lines, coordinate geometry (also known as the analytic geometry) explains how geometry and algebra are related. They can answer geometrical problems because it gives them geometrical aspects of algebra.

Collinear means a straight line

For three points to collinear, the are of triangle must be zero

Area of triangle=½∣x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)∣

Given:

the point (1,-1),(k,1) and (4,5) are collinear

Find:

The value of k for which the poins are collinear

Solution:

Consider the given points.

(1,−1),(k,1) and (4,5)

Since, these points are collinear means that the area of triangle must me zero.,

½∣x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)∣=0

where (x₁,y₁),(x₂,y₂),(x₃,y₃) are the points

Therefore,

⇒1(1−5)+k(5+1)+4(−1−1)=0

⇒1(−4)+k(6)+4(−2)=0

⇒−4+12k−8=0

⇒−4+4K=0

⇒4k=4

⇒k=1

Therefore, for k=1 the point (1,-1),(k,1) and (4,5) are collinear

#SPJ3