If (a, 0) , (0, b) and (x, y) are collinear, thena)ay – bx = 1b)ay + bx = abc)ax – by = abd)ax + by = 1Correct answer is option 'B'. Can you explain this answer?

Question

If (a, 0) , (0, b) and (x, y) are collinear, thena)ay &#8211; bx = 1b)ay + bx = abc)ax &#8211; by = abd)ax + by = 1Correct answer is option 'B'. Can you explain this answer?

Answer

Let x1=a , y1=0 , x2=0 , y2=b, x3=x , y3=y For collinear,                         x1(y2-y3)+x2(y3-y1)+x3(y1-y2)  = 0                  a(b - y) + 0 (y  -  0)  + x  (0-b )   =0                    ab - ay + 0 + 0 - bx  =0                   ab = ay + bx                   ay + bx  = ab.

Answer

Explanation:

If given points are collinear, the area of triangle formed by these three points is 0.

Answer

Given information
- Three points (a, 0), (0, b) and (x, y) are collinear.
- Collinear points lie on the same line.

To prove
ay - bx = 0

Proof
Let's assume that the three points are collinear, and they lie on a line with equation y = mx + c, where m is the slope of the line and c is the y-intercept.

- (a, 0) lies on the line, so we have 0 = ma + c or c = -ma.
- (0, b) lies on the line, so we have b = 0 + c or c = b.
- (x, y) lies on the line, so we have y = mx + c.

Substituting c = -ma from the first equation into the second equation, we get b = -ma, or m = -b/a.

Substituting m = -b/a and c = b into the equation for the line, we get y = (-b/a)x + b or ay + bx = ab.

Multiplying both sides of the equation by -1, we get -ay - bx = -ab.

Adding this equation to the previous equation, we get ay - bx = 0, which is the desired result.

Therefore, option B is the correct answer.

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