Question

The equation of the line \[x + y = 2\] in double intercept form, the \[x - \]intercept, and the \[y - \]intercept, respectively are
(a) \[\dfrac{x}{2} - \dfrac{y}{2} = 1\], \[a = 2\], and \[b = - 2\]
(b) \[ - \dfrac{x}{2} + \dfrac{y}{2} = 1\], \[a = - 2\], and \[b = 2\]
(c) \[\dfrac{x}{2} + \dfrac{y}{2} = 1\], \[a = 2\], and \[b = 2\]
(d) \[ - \dfrac{x}{2} - \dfrac{y}{2} = 1\], \[a = - 2\], and \[b = - 2\]

Answer 1

Hint:
Here, we will rewrite the given equation to get the equation in double intercept form. Then, we will use the formula for the equation of the line in double intercept form to find the values of the \[x - \] intercept and the \[y - \] intercept of the line.
Formula Used:
The double intercept form of a line is given by the equation \[\dfrac{x}{a} + \dfrac{y}{b} = 1\], where \[a\] is the \[x - \] intercept of the line and \[b\] is the \[y - \] intercept of the line.

Complete step by step solution:
We will rewrite the given equation of the line in its double intercept form.
Dividing both sides of the equation \[x + y = 2\] by 2, we get
\[ \Rightarrow \dfrac{{x + y}}{2} = \dfrac{2}{2}\]
Simplifying the equation, we get
\[ \Rightarrow \dfrac{{x + y}}{2} = 1\]
Rewriting the equation by splitting the L.C.M., we get
\[ \Rightarrow \dfrac{x}{2} + \dfrac{y}{2} = 1\]
We can observe that this is the double intercept form of the given line.
Now, we will find the values of the \[x - \]intercept and the \[y - \]intercept of the line.
Comparing the double intercept form of the line \[\dfrac{x}{2} + \dfrac{y}{2} = 1\] to \[\dfrac{x}{a} + \dfrac{y}{b} = 1\], we get
\[a = 2\] and \[b = 2\].
\[\therefore \] We get the \[x -\] intercept and the \[y -\] intercept of the line as \[a = 2\] and \[b = 2\] respectively.

Thus, the correct option is option (c).

Note:
The \[x - \]intercept of a line is the point at which it touches the \[x - \]axis. It is given by the point \[\left( {a,0} \right)\]. Similarly, the \[y - \]intercept of a line is the point at which it touches the \[y - \]axis. It is given by the point \[\left( {0,b} \right)\]. We can easily find the \[y - \]intercept by substituting \[x = 0\] in the given equation.
\[\begin{array}{l}x + y = 2\\ \Rightarrow 0 + y = 2\\ \Rightarrow y = 2\end{array}\]
So the \[y - \]intercept of the line is \[\left( {0,2} \right)\].
Now we can find \[x - \]intercept by substituting \[y = 0\] in the given equation.
\[\begin{array}{l}x + y = 2\\ \Rightarrow x + 0 = 2\\ \Rightarrow x = 2\end{array}\]
So the \[x - \]intercept of the line is \[\left( {2,0} \right)\].