If the point 𝑃 at 𝑥, 𝑦 is
equidistant from the points 𝐴 at 𝑎 plus 𝑏, 𝑏 minus 𝑎 and 𝐵 at 𝑎 minus 𝑏, 𝑎
plus 𝑏, prove that 𝑏𝑥 is equal to 𝑎𝑦.
We’ll need to apply the distance
formula here. The distance formula is derived
from the Pythagorean theorem. And it says that for two points at
𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two, the distance between them is given by the square
root of the square of the differences between the 𝑥-coordinates plus the square of
the differences between the 𝑦-coordinates.
We can find the distance of the
lines joining 𝐴 and 𝑃 and 𝐵 and 𝑃 by substituting the relevant values into this
formula. For 𝐴𝑃, let’s call 𝑥, 𝑦 𝑥 one,
𝑦 one and the point at 𝐴 𝑥 two, 𝑦 two. We can then substitute these values
into the formula as shown. The difference between the
𝑥-coordinates is 𝑎 plus 𝑏 minus 𝑥. And the difference between the
𝑦-coordinates is 𝑏 minus 𝑎 minus 𝑦.
We can repeat this process for the
distance between the points 𝑃 and 𝐵. The difference between the
𝑥-coordinates is 𝑎 minus 𝑏 minus 𝑥. And the difference between the
𝑦-coordinates is 𝑎 plus 𝑏 minus 𝑦. Since the point 𝑃 is equidistant
from the points 𝐴 and 𝐵, we know that the distance between 𝐴 and 𝑃 is equal to
the distance between 𝐵 and 𝑃. This means we can equate the two
expressions we formed for the distance between the lines.
At this point, we can square both
sides of this equation. We will, however, need to expand
each of these individual brackets. This is much like expanding
products of binomials. We’ll need to make sure that each
term in the first bracket multiplies by each term in the second. We can use a grid method though to
do this and ensure we don’t lose any terms. Let’s multiply out this first pair
Multiplying each term on the top of
the grid by each term on the side, and we end up with these individual terms. We can repeat this for the second
pair of brackets. And we get these terms. Remember, these are being
added. So let’s add together everything we
have in these two grids.
𝑎𝑏 minus 𝑎𝑏 is zero. So these 𝑎𝑏 terms cancel out. Adding these all together and we’re
left with a rather nasty looking expression. But these should cancel out later
on down the line. Multiplying each pair of brackets
on the right-hand side of the equation, and we get this. Once again, we’re finding the sum
of these expressions. So the 𝑎𝑏s cancel out. And when we simplify by collecting
like terms, we get this.
Notice, we can subtract two 𝑎
squared and two 𝑏 squared from both sides. We can add two 𝑎𝑥 to both
sides. We can add two 𝑏𝑦 to both
sides. And we can subtract an 𝑥 squared
and a 𝑦 squared. And this is what we’re left
with. This looks much nicer at this
To simplify further, we’ll add two
𝑏𝑥 to both sides of the equation. We get two 𝑎𝑦 equals four 𝑏𝑥
minus two 𝑎𝑦. Next, we’ll add two 𝑎𝑦 to both
sides of the equation, to leave us with four 𝑎𝑦 equals four 𝑏𝑥. We could divide everything by
four. And changing the order to match the
question, we’ve proved that 𝑏𝑥 is equal to 𝑎𝑦.