Confused regarding the answer of a problem based on locus.

Question

I have a question on locus which goes like this. $A(5,3)$ and $B(3,-2)$ are two fixed points. Find the equation of the locus of $P$, so that the triangle $PAB$ is 9. Now the loci of the point $P$ should be 2 parallel lines on either side of $AB$. So we get equations of two lines. One is $5x-2y-37$ and other is $5x-2y-1$. The answer in answer key is the locus of $P$ lies on $(5x-2y-37)(5x-2y-1)=0$ But I don't understand why shouldn't we present the answer as, the loci of point P are lines with equations $5x-2y-37$ and $5x-2y-1$. Why do we write them as factors? Hope my question is clear.

Answer 1

What you have done is almost correct except you should say "one is $5x-2y-37 \bf= 0$ and ....".

You have the right to give your answer in two separate equations or equivalently in the way presented in the answer key. It is a form called system of equations or pair of equations. It bundled two equations together.

Answer 2

A locus represents the set of points that fulfill a specific condition, often expressed using a so-called implicit equation like $$f(x,y)=0.$$

If your analysis reveals that there are two separate cases, you can express the locus with a logical disjunction, as

$$f(x,y)=0\lor g(x,y)=0.$$

But this is equivalent to saying that the product is zero:

$$f(x,y)\cdot g(x,y)=0,$$

which is also of the form $$h(x,y)=0.$$

Mostly a matter of taste.

Answer 3

$(5x−2y−37)(5x−2y−1)=0 \quad \iff \quad 5x−2y−37=0 \quad {\rm or} \quad 5x−2y−1=0$.

The former equation is a more compact, albeit cute, way of expressing the answer.