Question

By using the concept of equation of a line, prove that the three points (-2, -2), (8, 2) and (3, 0) are collinear.

Answer 1

$\large\underline{\sf{Solution-}}$

We have to prove that, three points (-2, -2), (8, 2) and (3, 0) are collinear using concept of equation of line.

Let assume that three points (-2, -2), (8, 2) and (3, 0) are represented as A, B and C.

Let first find the equation of line AB which passes through the point (- 2, - 2) and (8, 2).

We know,

Two point form of equation of line :- The equation of line which passes through the points A(x₁, y₁) and B(x₂, y₂) is given by

$\boxed{\rm{  \:y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) \: \: }}$

So, on substituting the values, we get

$\rm \: y + 2 = \dfrac{2 + 2}{8 + 2}(x + 2)$

$\rm \: y + 2 = \dfrac{4}{10}(x + 2)$

$\rm \: y + 2 = \dfrac{2}{5}(x + 2)$

$\rm \: 5y + 10 = 2x + 4$

$\rm\implies \:2x - 5y = 6$

Now, we have to check whether point C(3, 0) lies on it.

So, on substituting x = 3 and y = 0, in above equation, we get

$\rm \: 2 \times 3 - 5 \times 0 = 6$

$\rm \: 6 - 0= 6$

$\rm \: 6= 6$

Hence, point C lies on AB.

Hence, three points (-2, -2), (8, 2) and (3, 0) are collinear.

$\rule{190pt}{2pt}$

Additional Information :-

Different forms of equations of a straight line

1. Equations of horizontal and vertical lines

Equation of line parallel to x - axis passes through the point (a, b) is y = b.

Equation of line parallel to y - axis passes through the point (a, b) is x = a.

2. Point-slope form equation of line

Equation of line passing through the point (a, b) having slope m is y - b = m(x - a)

3. Slope-intercept form equation of line

Equation of line which makes an intercept of c units on y axis and having slope m is y = mx + c.

4. Intercept Form of Line

Equation of line which makes an intercept of a and b units on x - axis and y - axis respectively is x/a + y/b = 1.

5. Normal form of Line

Equation of line which is at a distance of p units from the origin and perpendicular makes an angle β with the positive X-axis is x cosβ + y sinβ = p.

Answer 2

CONCEPT AND TIP :-

• Three or more points are said to be collinear if they lie on the same line and the points which do not lie on the same line are known as non-collinear points.

For example:

• In the figure given below,

• the points A, B, C are collinear points since they lie on the same line and D, E, F are non collinear points as they don't lie on single line if drawn.

USING CONCEPT :-

• formula to find collinear point
• m = y2 - y1 / x 2 - x1

TO Prove :-

• prove that the three points (-2, -2), (8, 2) and (3, 0) are collinear.

SOLUTION :-

The slope of the line passing through

thepoints (-2,-2) and (8,2) is

• Y2 - Y1 / x 2 - x1 = 2 +2 / 8 + 2 = 4/10 = 2/5

The equation of the line passing through the points (-2,-2) and (8,2) is

• y - y₁= m(x - x₁)

• y + 2 = 2/5 ( x +2)

• 5y + 10 = 2x + 4

• 2x - 5y - 6= 0

Clearly (3,0) satisfies the equation and

the line passing through the points (-2,-2) and

(8,2) also passes through (3,0) Hence,

the three points are collinear.