Question

If (x,y) is any point on a straight line which passes through (a,0) and (b,0) , prove that x/a + y/b = 1

Answer 1

Appropriate Question :-

If (x,y) is any point on a straight line which passes through (a,0) and (0,b) , prove that x/a + y/b = 1

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

Given that,

• (x,y) is any point on a straight line which passes through (a,0) and (0,b).

Let assume that

• (a, 0) represents the coordinates of A.

• (0, b) represents the coordinates of B.

• (x, y) represents the coordinates of P.

It implies, points A, B and P are collinear.

We know, Slope of line joining the points (a, b) and (c, d) is given by

$\boxed{ \tt{ \: Slope = \frac{d - b}{c - a} \: }}$

We know, 3 points A, B and P are collinear iff

• Slope of AP = Slope of BP

THUS,

$\rm :\longmapsto\:\dfrac{y - 0}{x - a} = \dfrac{y - b}{x - 0}$

$\rm :\longmapsto\:\dfrac{y}{x - a} = \dfrac{y - b}{x}$

$\rm :\longmapsto\:xy = (x - a)(y - b)$

$\rm :\longmapsto\:xy = xy - bx - ay + ab$

$\rm :\longmapsto\:0 = - bx - ay + ab$

$\rm :\longmapsto\:bx + ay = ab$

On dividing both sides by ab, we get

$\rm :\longmapsto\:\dfrac{bx}{ab} + \dfrac{ay}{ab} = \dfrac{ab}{ab}$

$\bf :\longmapsto\:\dfrac{x}{a} + \dfrac{y}{b} =1$

Hence, Proved

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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 x = a.

Equation of line parallel to x - 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.