Question

Find the equation. of line such that the sum of its intercepts is o and it passes through A (4,1)​

Answer 1

Let the line make intercepts of d and c on x and y axes respectively.

Then by intercept form, the equation of the line will be,

$\longrightarrow\dfrac{x}{d}+\dfrac{y}{c}=1$

Given that the sum of the intercepts is zero, i.e.,

$\longrightarrow c+d=0$

$\longrightarrow c=-d$

Then equation of the line becomes,

$\longrightarrow\dfrac{x}{d}+\dfrac{y}{-d}=1$

$\longrightarrow\dfrac{x}{d}-\dfrac{y}{d}=1$

$\longrightarrow x-y=d$

Given that the line passes through (4, 1).

Taking (x, y) = (4, 1),

$\longrightarrow 4-1=d$

$\longrightarrow d=3$

Hence equation of the line is,

$\longrightarrow\underline{\underline{x-y=3}}$

Answer 2

$\sf \large \underline{ \green{Given: - }}$

• Sum of the intercept of line is 0

• Line Passes through the point A(4, 1)

$\sf\large\underline\purple{To\: Find:-}$

• Equation of line

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

To find the equation of line which passes through the point A(4, 1) and such that sum of the intercept on the axes is 0, is evaluated by using the formula of Intercept Form of a line.

$\sf\large\underline\purple{Formula\:used:-}$

Let us assume a line which makes an intercept of 'a' and 'b' units on axes, then equation of line using intercept form a line is given by

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

Now,

$\sf\underline\purple{Let \: assume \: that }$

• Intercept on x - axis = a

• Intercept on y - axis = b

So,

• Equation of line is given by

$\tt{ \implies{ \green{\dfrac{x}{a} + \dfrac{y}{b} = 1}}} - - - (1)$

Since,

• Sum of the intercept on the axes is 0.

$\tt{ \implies \: {a \: + \: b \: = \: 0}}$

$\tt{ \implies \: {b \: = \: - \: a}}$

So,

• Equation (1) can be rewritten as

$\tt{ \implies{{\dfrac{x}{a} + \dfrac{y}{ - a} = 1}}}$

$\tt{ \implies{ {\dfrac{x}{a} - \dfrac{y}{a} = 1}}}$

$\tt{ \implies \: x - y \: = \: a} - - (2)$

Now,

• Equation (2) passes through the point A(4, 1), So

$\tt{ \implies \: 4 \: - \: 1 \: = \: a}$

$\boxed{ \pink{ \tt{ \implies \: a \: = \: 3}}}$

Substituting this value of 'a' in Equation (2), we get

$\sf\large\underline\purple{ \implies \: {x \: - \: y \: = \: 3\: }}$

Additional Information

Additional Information about Different forms of equations of a straight line

1. Equations of horizontal and vertical lines

• Equation of the lines which are horizontal or parallel to the X-axis is y = a, where a is the y – coordinate of the points on the line.
• Similarly, equation of a straight line which is vertical or parallel to Y-axis is x = a, where a is the x-coordinate of the points on the line.

2. Point-slope form equation of line

• Consider a non-vertical line L whose slope is m, A(x,y) be an arbitrary point on the line and P(a, b) be the fixed point on the same line. Equation of line is given by y - b = m(x - a)

3. Slope-intercept form equation of line

• Consider a line whose slope is m which cuts the Y-axis at a distance ‘a’ from the origin. Then the distance a is called the y– intercept of the line. The point at which the line cuts y-axis will be (0,a). Then equation of line is given by y = mx + a.

4. Intercept Form of Line

• Consider a line L having x– intercept a and y– intercept b, then the line passes through  X– axis at (a,0) and Y– axis at (0,b).
• Equation of line is given by x/a + y/b = 1.

5. Normal form of Line

• Consider a perpendicular from the origin having length p to line L and it makes an angle β with the positive X-axis. Then, equation of line is given by
• x cosβ + y sinβ = p.