For what values of $a$ and $b$ the intercepts 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.

For what values of a and b the intercepts cut off on the coordinate axes by the line ax+by+8=0 are equal in le

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Concept understanding — Coordinate Axes Equations

Coordinate Axes Equations

Imagine a blank sheet of paper. You want to describe exactly where a point is. The most natural way is to draw two perpendicular lines — one horizontal, one vertical — and measure distances from them. Those two lines are the coordinate axes.

The horizontal line is the x-axis. The vertical line is the y-axis. Where they cross is the origin, the point $(0,0)$ .

Now, here's the key idea: every point on the x-axis has a y-coordinate of 0. Why? Because to be on that horizontal line, you haven't moved up or down at all — your vertical distance from the axis is zero. Similarly, every point on the y-axis has an x-coordinate of 0 — you haven't moved left or right from the vertical line.

That's the entire intuition. Let's make it precise.

The Equations

\text{x-axis: } y = 0

That's it. These are the equations of the coordinate axes.

$y = 0$ means: "all points where the y-coordinate is zero, regardless of x." That's the entire horizontal line through the origin.
$x = 0$ means: "all points where the x-coordinate is zero, regardless of y." That's the entire vertical line through the origin.

Watch out

A common mistake is to think $y=0$ means "the point $(0,0)$ ". No — it means every point with $y=0$ , like $(5,0)$ , $(-3,0)$ , $(100,0)$ , etc. It's a whole line, not a single point.

Why This Matters

These two equations are the foundation for everything in coordinate geometry. Every other line, curve, or shape is described relative to these axes. For example:

The line $y = 2$ is a horizontal line parallel to the x-axis, shifted up by 2 units.
The line $x = -3$ is a vertical line parallel to the y-axis, shifted left by 3 units.

So when you see $y = 0$ or $x = 0$ , think: "the original, unshifted axes themselves." …

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Converting both lines to intercept form, $2x-3y+6=0$ has intercepts $-3$ and $2$ . The line $ax+by+8=0$ must have intercepts $3$ and $-2$ (equal in length, opposite in sign). Matching these gives $a=-\dfrac{8}{3}$ and $b=4$ .

Setting up: intercept form

A line $Ax+By+C=0$ (with $C\neq 0$ ) can be rewritten in intercept form:

\frac{x}{X}+\frac{y}{Y}=1

where $X$ is the x-intercept and $Y$ is the y-intercept, found by dividing the whole equation by $-C$ .

Step 1: Intercepts of $2x-3y+6=0$

2x-3y+6=0 \implies 2x-3y=-6

Divide by $-6$ :

\frac{2x}{-6}-\frac{3y}{-6}=1 \implies \frac{x}{-3}+\frac{y}{2}=1

So this line's intercepts are $X_1=-3$ and $Y_1=2$ .

Step 2: Required intercepts of $ax+by+8=0$

The intercepts of $ax+by+8=0$ must be equal in length but opposite in sign to $X_1,Y_1$ : …

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Q.For what values of aaa and bbb the intercepts cut off on the coordinate axes by the line ax+by+8=0ax+by+8=0ax+by+8=0 are equal in length but opposite in signs to those cut off by the line 2x−3y+6=02x-3y+6=02x−3y+6=0 on the axes.

Setting up: intercept form

Step 1: Intercepts of 2x−3y+6=02x-3y+6=02x−3y+6=0

Step 2: Required intercepts of ax+by+8=0ax+by+8=0ax+by+8=0

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Q.For what values of $a$ and $b$ the intercepts 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.

Step 1: Intercepts of $2x-3y+6=0$

Step 2: Required intercepts of $ax+by+8=0$

Unlock everything free for 14 days