Question

The area of the triangle formed by the line
x/a+y/b=1 with the coordinate axes is
​

Answer 1

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

Given equation of line is

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

Let assume that the given line intersects the x - axis at A and intersects the y - axis at B.

So, we have to find the area of triangle AOB.

Now, let first evaluate the coordinates of A and B.

Point of intersection of line with x - axis

We know, on x - axis, y = 0

So, given equation can be rewritten as

$\rm \: \dfrac{x}{a} + \dfrac{0}{b} = 1$

$\rm \: \dfrac{x}{a} + 0 = 1$

$\rm \: \dfrac{x}{a} = 1$

$\rm\implies \:x = a$

So, Coordinates of A is (a, 0).

So, OA = a units

Now, Point of intersection with y - axis

We know, on y - axis, x = 0

$\rm \: \dfrac{0}{a} + \dfrac{y}{b} = 1$

$\rm \: 0 + \dfrac{y}{b} = 1$

$\rm \: \dfrac{y}{b} = 1$

$\rm\implies \:y = b$

So, Coordinates of B is (0, b).

So, OB = b units

So, we have In right triangle AOB, right angled at O,

OA = a units

OB = b units

$\rm \: Area_{(\triangle\:AOB)} = \dfrac{1}{2} \times OA \times OB$

$\rm \: Area_{(\triangle\:AOB)} = \dfrac{1}{2} \times a \times b$

$\rm\implies \:Area_{\triangle\:AOB} = \dfrac{ab}{2} \: square \: units$

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Short Cut Trick

The area bounded between the coordinates axis by the line ax + by + c = 0 is

$\boxed{\sf{ \: \: Area_{(\triangle)} = \frac{ {c}^{2} }{2 |ab| } \: \: }}$

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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 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.