Find the equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9.
Solution and Explanation
The equation of a line in the intercept form is
\(\frac{x}{y}+\frac{y}{b}=1\) \(......(i)\)
Here, a and b are the intercepts on x and y axes respectively.
It is given that
\(a + b = 9 ⇒ b = 9a \,\,\,\, … (ii)\)
From equations (i) and (ii), we obtain
\(\frac{x}{a}+\frac{y}{9-a}=1\) \(......(iii)\)
It is given that the line passes through point (2, 2). Therefore, equation (iii) reduces to
\(\frac{2}{a}+\frac{2}{9-a}=1\)
\(⇒2(\frac{1}{a}+\frac{1}{9-a})=1\)
\(⇒2(\frac{9-a+a}{a(9-a)}=1\)
\(⇒\frac{18}{9a-a^2}=1\)
\(⇒18=9a-a^2\)
\(⇒a^2-9a+18=0\)
\(⇒a^2-6a-3a+18=0\)
\(⇒a(a-6)-3(a-6)=0\)
\(⇒(a-6)(a-3)=0\)
\(⇒a=6\,or\,a=3\)
If a = 6 and b = 9 -6 = 3, then the equation of the line is
\(\frac{x}{6}+\frac{y}{3}=1⇒x+2y-6=0\)
If a = 3 and b = 9-3 = 6, then the equation of the line is
\(\frac{x}{3}+\frac{y}{6}=1⇒2x+y-6=0\)
Top Questions on Various Forms of the Equation of a Line
- A line passes through (2, 2) and is perpendicular to the line 3x + y = 3. Its y-intercept is
- KCET - 2023
- Mathematics
- Various Forms of the Equation of a Line
- Suppose $ P(2,\,y,\,z) $ lies on the line through $ A(3,-1,4) $ and $ B(-4,2,1) $ . Then, the value of z is equal to
- KCET - 2011
- Mathematics
- Various Forms of the Equation of a Line
- $(0, - 1)$ and $(0, 3)$ are two opposite vertices of a square. The other two vertices are :
- BITSAT - 2006
- Mathematics
- Various Forms of the Equation of a Line
- If lines \(\frac{x-1}{-3}=\frac{y-2}{2k}=\frac{z-3}{2}\) and \(\frac{x-1}{3k}=\frac{y-5}{1}=\frac{z-6}{-5}\) are mutually perpendicular then k is equal to
- KCET
- Mathematics
- Various Forms of the Equation of a Line
- A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio 1:n. Find the equation of the line.
- CBSE Class XI
- Mathematics
- Various Forms of the Equation of a Line
Questions Asked in CBSE Class XI exam
- Distinguish between the following pairs of sentences.
(i) He was visibly moved.
(ii) He was visually impaired.- CBSE Class XI
- The adventure
- How, according to you, can peace and liberty be maintained in a state?
- CBSE Class XI
- The tale of melon City
Figures 9.20(a) and (b) refer to the steady flow of a (non-viscous) liquid. Which of the two figures is incorrect ? Why ?
- CBSE Class XI
- Pressure
- Describe the change in hybridisation (if any) of the Al atom in the following reaction.
\(AlCl_3+Cl^- \rightarrow AlCl_4^-\)- CBSE Class XI
- Hybridisation
- Why did the narrator of the story want to forget the address?
- CBSE Class XI
- The address
Concepts Used:
Straight lines
A straight line is a line having the shortest distance between two points.
A straight line can be represented as an equation in various forms, as show in the image below:
The following are the many forms of the equation of the line that are presented in straight line-
1. Slope – Point Form
Assume P0(x0, y0) is a fixed point on a non-vertical line L with m as its slope. If P (x, y) is an arbitrary point on L, then the point (x, y) lies on the line with slope m through the fixed point (x0, y0) if and only if its coordinates fulfil the equation below.
y – y0 = m (x – x0)
2. Two – Point Form
Let's look at the line. L crosses between two places. P1(x1, y1) and P2(x2, y2) are general points on L, while P (x, y) is a general point on L. As a result, the three points P1, P2, and P are collinear, and it becomes
The slope of P2P = The slope of P1P2 , i.e.
\(\frac{y-y_1}{x-x_1} = \frac{y_2-y_1}{x_2-x_1}\)
Hence, the equation becomes:
y - y1 =\( \frac{y_2-y_1}{x_2-x_1} (x-x1)\)
3. Slope-Intercept Form
Assume that a line L with slope m intersects the y-axis at a distance c from the origin, and that the distance c is referred to as the line L's y-intercept. As a result, the coordinates of the spot on the y-axis where the line intersects are (0, c). As a result, the slope of the line L is m, and it passes through a fixed point (0, c). The equation of the line L thus obtained from the slope – point form is given by
y – c =m( x - 0 )
As a result, the point (x, y) on the line with slope m and y-intercept c lies on the line, if and only if
y = m x +c