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For what values of a and b the intercept cut off on the coordinate axes by the line ax + by + 8 = 0 are equal in length but opposite in signs to those cut off by the line 2x – 3y + 6 = 0 on the axes.

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Hint: For a line having intercept a on the x – axis and intercept b on the y – axis, the equation of this line is given by $\dfrac{x}{a}+\dfrac{y}{b}=1$. This form of line is also called the intercept form of the line. Any line can be expressed in this form and then we can find the intercept of the line on x and y axes. Using this form of line, we can solve this question.

Complete step-by-step answer:

Before proceeding with the question, we must know the formula that will be required to solve this question.
In coordinate geometry, the equation of the straight line having its x intercept equal to a and y intercept equal to b is given by,
$\dfrac{x}{a}+\dfrac{y}{b}=1$ . . . . . . . . . . . . . (1)
In the question, it is given a line ax + by + 8 = 0. It is given that this line has both the intercepts equal in length. Let us try to express this line in intercept form.
\[\begin{align}
  & ax+by+8=0 \\
 & \Rightarrow ax+by=-8 \\
 & \Rightarrow \dfrac{ax}{-8}+\dfrac{by}{-8}=1 \\
 & \Rightarrow \dfrac{x}{\dfrac{-8}{a}}+\dfrac{y}{\dfrac{-8}{b}}=1 \\
 & \\
\end{align}\]
From (1), we can say that x intercept of this line is \[\dfrac{-8}{a}\] and y intercept of this line is \[\dfrac{-8}{b}\].
Also, in the question, we are given a line 2x – 3y + 6 = 0. Let us find the intercepts of this line by first converting it to intercept form.
$\begin{align}
  & 2x-3y+6=0 \\
 & \Rightarrow 2x-3y=-6 \\
 & \Rightarrow \dfrac{2x}{-6}-\dfrac{3y}{-6}=1 \\
 & \Rightarrow \dfrac{x}{-3}+\dfrac{y}{2}=1 \\
\end{align}$
So, from (1), we can say that the x intercept of this line is -3 and y intercept of this line is 2.
In the question, it is given that the intercepts of the line ax + by + 8 = 0 are equal in length but opposite in signs to those cut off by the line 2x – 3y + 6 = 0 on the axes. So, the x intercept of line ax + by + 8 = 0 is +3 and the y intercept of the line ax + by + 8 = 0 is -2. Also, the x intercept of line ax + by + 8 = 0 is $\dfrac{-8}{a}$ and the y intercept of the line ax + by + 8 = 0 is $\dfrac{-8}{b}$. So, we get,
$\dfrac{-8}{a}=3$ and $\dfrac{-8}{b}=-2$
$\Rightarrow a=\dfrac{-8}{3}$ and $b=4$.

Note: The intercepts of any line can also be found by substituting x = 0 and y = 0. Substituting x = 0 and solving for y gives the y intercept of the line and substituting y = 0 and solving for x gives the x intercept of the line.

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