Maths > Straight Lines > 3.0 Equation of a straight line in various forms
Straight Lines
1.0 Definition
2.0 Condition of collinearity of three points
3.0 Equation of a straight line in various forms
3.1 Point-slope form
3.2 Slope-Intercept form
3.3 Two point form
3.4 Determinant form
3.5 Intercept form
3.6 Normal form
3.7 Parametric form
4.0 Angle between two lines
5.0 Length of perpendicular from a point to a line
6.0 Foot of perpendicular from a point to a line
7.0 Reflection of a point about a line
7.1 Reflection with respect to $X-$axis
7.2 Reflection with respect to $Y-$axis
7.3 Reflection with respect to Origin
7.4 Reflection with respect to the line $x=a$
7.5 Reflection with respect to the line $y=b$
7.6 Reflection with respect to the line $y=x$
8.0 Equation of angle bisector
9.0 Bisector of angle containing origin
10.0 Bisector of angle containing a given point
11.0 Family of straight lines
3.5 Intercept form
3.2 Slope-Intercept form
3.3 Two point form
3.4 Determinant form
3.5 Intercept form
3.6 Normal form
3.7 Parametric form
7.2 Reflection with respect to $Y-$axis
7.3 Reflection with respect to Origin
7.4 Reflection with respect to the line $x=a$
7.5 Reflection with respect to the line $y=b$
7.6 Reflection with respect to the line $y=x$
The equation of straight line which makes intercept of $a$ and $b$ on $X-$axis and $Y-$axis respectively is written as
$$\frac{x}{a} + \frac{y}{b} = 1$$
Question 2. Find the equation of a line which passes through the point $(3,4)$ and the sum of its intercepts on the axes is $14$.
Solution: The equation of straight line in Intercept form is $\frac{x}{a} + \frac{y}{b} = 1$ which passes through $(3,4)$. Therefore, $$\frac{3}{a} + \frac{4}{b} = 1...(1)$$
and sum of intercepts i.e., $$a+b=14...(2)$$
Put value of $b$ from $(2)$ in $(1)$, we get $$\begin{equation} \begin{aligned} {a^2} - 13a + 42 = 0 \\ (a - 7)(a - 6) = 0 \\ a = 7{\text{ or }}a = 6 \\\end{aligned} \end{equation} $$
For $a=7$, $b=14-7=7$ and
For $a=6$, $b=14-6=8$
Putting the values of $a$ and $b$ in the equation of line in intercept form, we get $$\begin{equation} \begin{aligned} \frac{x}{7} + \frac{y}{7} = 1{\text{ and }}\frac{x}{6} + \frac{y}{8} = 1 \\ or \\ x + y = 7{\text{ and }}4x + 3y = 24 \\\end{aligned} \end{equation} $$
