I understand that the point will be at the midpoint, but the answer says smthing else
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Find a point on the Y axis which is equidistant from the points A(6,5) and B(-4, 3).
I understand that the point will be at the midpoint, but the answer says smthing else
I understand that the point will be at the midpoint, but the answer says smthing else
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Brent@GMATPrepNow
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We'll use the following formula:[email protected] wrote:Find a point on the Y axis which is equidistant from the points A(6,5) and B(-4, 3).
The distance between the points (a,b) and (c,d) = sqrt[(a-c)² + (b-d)²]
IMPORTANT: if we're looking for a point on the y-axis, then the x-coordinate of that point will equal 0. So, let's say that (0,y) is the point that's equidistant from points A and B.
We want: the distance from point A to (0,y) = the distance from point B to (0,y)
In other words, the distance from (6,5) to (0,y) = the distance from (-4, 3) to (0,y)
Apply formula: sqrt[(6 - 0)² + (5 - y)²] = sqrt[(-4 - 0)² + (3 - y)²]
Square both sides: (6 - 0)² + (5 - y)² = (-4 - 0)² + (3 - y)²
Expand and simplify: 36 + 25 - 10y + y² = 16 + 9 - 6y + y²
Simplify: 61 - 10y + y² = 25 - 6y + y²
Simplify: 36 = 4y
Solve: y = 9
So, [spoiler](0, 9)[/spoiler] is the point on the y-axis that's equidistant from the points A(6,5) and B(-4, 3).
Cheers,
Brent
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But Brent, the point is equidistant doesn't it imply that the point is in the middle of both Y and X,
So cant we use the midpoint formula for this?
So cant we use the midpoint formula for this?
Brent@GMATPrepNow wrote:We'll use the following formula:[email protected] wrote:Find a point on the Y axis which is equidistant from the points A(6,5) and B(-4, 3).
The distance between the points (a,b) and (c,d) = sqrt[(a-c)² + (b-d)²]
IMPORTANT: if we're looking for a point on the y-axis, then the x-coordinate of that point will equal 0. So, let's say that (0,y) is the point that's equidistant from points A and B.
We want: the distance from point A to (0,y) = the distance from point B to (0,y)
In other words, the distance from (6,5) to (0,y) = the distance from (-4, 3) to (0,y)
Apply formula: sqrt[(6 - 0)² + (5 - y)²] = sqrt[(-4 - 0)² + (3 - y)²]
Square both sides: (6 - 0)² + (5 - y)² = (-4 - 0)² + (3 - y)²
Expand and simplify: 36 + 25 - 10y + y² = 16 + 9 - 6y + y²
Simplify: 61 - 10y + y² = 25 - 6y + y²
Simplify: 36 = 4y
Solve: y = 9
So, [spoiler](0, 9)[/spoiler] is the point on the y-axis that's equidistant from the points A(6,5) and B(-4, 3).
Cheers,
Brent
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Brent@GMATPrepNow
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The midpoint between A and B will, indeed, be equidistant from A and B. However, the midpoint will not be on the y-axis, and the question tells us that the point is on the y-axis.[email protected] wrote:But Brent, the point is equidistant doesn't it imply that the point is in the middle of both Y and X,
So cant we use the midpoint formula for this?
The truth is that there is an infinite number of points that are equidistant from the points A(6,5) and B(-4, 3). To get a better idea of what I mean, check out this related post: https://www.beatthegmat.com/dint-get-the ... 67535.html
Cheers,
Brent
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mgm
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Brent I did get the same answer but then I thought abt an alternative option as well: the point is below the x axis.
So (y+5)^2 + 36 = (y+3)^2 + 16
You get a different solution
Where is this problem from?
So (y+5)^2 + 36 = (y+3)^2 + 16
You get a different solution
Where is this problem from?
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Brent@GMATPrepNow
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How did you get (y+5) and (y+3)?mgm wrote:Brent I did get the same answer but then I thought abt an alternative option as well: the point is below the x axis.
So (y+5)^2 + 36 = (y+3)^2 + 16
You get a different solution
Where is this problem from?
We are finding the distance from (6,5) to (0,y) = the distance from (-4, 3) to (0,y)
So, the equation will be EITHER sqrt[(6 - 0)² + (5 - y)²] = sqrt[(-4 - 0)² + (3 - y)²]
OR sqrt[(0 - 6)² + (y - 5)²] = sqrt[(0 - -4)² + (y - 3)²]
Cheers,
Brent
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When we say that the point (0, y) is the point equidistant from points A and B, we are already allowing for the possibility that y may be negative. But it turns out that y is positive (y = 9)mgm wrote:Brent,
If you want to find a point below the X Axis that is equidistant from the two points and get the distance from say (0,-y) and the two points then you get a different answer...
Every point on the red line is equidistant from points A and B.
The question asks us to find a particular point on the red line. It asks us to find the point that's on the y-axis (where x = 0).
The coordinates of that point as (0, 9)
As you can see from the graph, there are no other points on the y-axis that are equidistant from A and B.
In your approach, you let (0, -y) be the equidistant point. All you're doing there is saying "let -y be the point on the y-axis that is equidistant from A and B." That's fine, BUT when you solve the equation, you must solve it for -y. When you do, you get -y = 9, which means (0, 9) is the equidistant point (great). HOWEVER, if you solve it for y, you get y = -9, which is not correct.
I hope that helps.
Cheers,
Brent
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