Implicit formula using equation in intercepts

Question

First of all I apologize if this is a trivial question, I am not sure if my solution is what the question is asking for.

Question :Using an equation in intercepts, obtain an implicit formula f(x,y)=0 for the straight line intersecting the coordinate axes X and Y at the points with coordinates (2, 0) and (0, -1), respectively

My thought process:

• Firstly, I will obtain M by subbing in the necessary values: (-1-0/0-2) = 1/2.
• Next, I will define the formula as y-y1 = M(x-x1), and then subbing in the values :y-0 = 1/2(x-2).
• Finally, I will turn it into implicit form : 2y - x + 2 = 0

May I have any guidance on whether what I did was correct? I am not sure if this was what I was supposed to do, I have also thought about x/a + y /b = 1, but I am not very sure how to get a and b in that instance

Answer 1

Making the ansatz $$y=mx+n$$ we get with $x=0$ directly $n=-1$.Now we obtain with $x=2$ and $y=0$ $$m=\frac{1}{2}$$

Answer 2

What you did looks correct, but you could have found an implicit formula for the line more directly.

Your thinking of $x/a+y/b=1$ was good.

Since the point where $x=2$ and $y=0$ is on the line, we see $2/a=1$; i.e., $a=2.$

And since the point where $x=0$ and $y=-1$ is on the line, we see $-1/b=1$, i.e., $b=-1.$

So right away we have the implicit form for the line: $x/2+y/(-1)=1$.

(You could multiply both sides by $2$ to get $x-2y=2$ or $x-2y-2=0$.)