Where does the point lies ?

[goelmohit2002]

Hi All,

The line segment PQ has the following coordinates:

P = (0,-1)
Q = (3,2)

the point on segment PQ that is twice as far from P as from Q is:

a) (3,1)
b) (2,1)
c) (2,-1)
d) (1.5, 0.5)
e) (1,0)

[spoiler]OA = B.[/spoiler]

Can someone please tell how to solve this question ? Using substitution becomes too lengthy.

Similarly using imaginary points ( x, y) too looks to be becoming very lengthy....

Thanks
Mohit

[truplayer256]

I really don't know any quicker way to solve this problem other than using this method:

Let the point on segment PQ be (x,y)

Distance from Q: sqrt[(x-3)^(2)+(y-2)^(2)]

Distance from P: 2*sqrt[(x-3)^(2)+(y-2)^(2)]

3*sqrt[(x-3)^(2)+(y-2)^(2)]=3*sqrt(2)

3*sqrt(2)= Distance from points (0,-1) and (3,2).

(x-3)^(2)+(y-2)^(2)=2

x^2+y^2-6x-4y=-11

Now test out the answer choices:

A.) (3,1)--> 9+1-18-4=-12 No.

B.) (2,1)--> 4+1-12-4=-11 Yes. This is our answer.

[goelmohit2002]

Hi truplayer256,

Thanks.

Basically what you are doing is substitution...for which I think we do not need to bother that much too....

Just simply find the distances from P and Q for every point....and compare with the data of the question....

But IMO that is just too much time taking...just think that points become awkward numbers and answer is option E

Especially it will become even more complex if this question comes in the DS....

is the data provided is sufficient to get the point in the question....

Is there any way to reach to the generalised way of solving this question...instead of using substitution.....

Thanks
Mohit

[Ian Stewart]

Hi All,

The line segment PQ has the following coordinates:

P = (0,-1)
Q = (3,2)

the point on segment PQ that is twice as far from P as from Q is:

For this particular question, you can make use of the answer choices. The midpoint is (1.5, 0.5), so we need a point which is on the line segment PQ and which is closer to Q than the midpoint. Only one answer choice makes sense.

To do any problem like this, remember how we find the midpoint between two points. If we have the points (3, 5) and (9,7), we just need to go halfway across from x=3, so to x = 6, and halfway up from y = 5, so to y = 6. That is, we divide the horizontal distance from 3 to 9 in a 1 to 1 ratio, and the vertical distance from 5 to 7 in a 1 to 1 ratio. That's equivalent to averaging the x and y co-ordinates, of course.

We can use the same principle no matter what the ratio. In the question above, we need a point that's twice as far from P = (0,-1) as from Q = (3,2). So we want to divide the horizontal distance from 0 to 3 in a 1 to 2 ratio, to get x = 2, and the vertical distance from -1 to 2 in a 1 to 2 ratio, to get y = 1. If you instead had the points P = (3, 11) and Q = (12, 5), we'd divide the distance from 3 to 12 in a 1 to 2 ratio, to get x = 9, and the distance from 11 to 5 in a 1 to 2 ratio, to get y = 7, and the point which is twice as far from P as from Q on the line PQ is (9,7).

[goelmohit2002]

We can use the same principle no matter what the ratio. In the question above, we need a point that's twice as far from P = (0,-1) as from Q = (3,2). So we want to divide the horizontal distance from 0 to 3 in a 1 to 2 ratio, to get x = 2, and the vertical distance from -1 to 2 in a 1 to 2 ratio, to get y = 1. If you instead had the points P = (3, 11) and Q = (12, 5), we'd divide the distance from 3 to 12 in a 1 to 2 ratio, to get x = 9, and the distance from 11 to 5 in a 1 to 2 ratio, to get y = 7, and the point which is twice as far from P as from Q on the line PQ is (9,7).

Awesome Ian....!! Thanks a lot.....

Kindly please tell.....the answer we shall get this way will always lie on the line segment only......although there is only one single point possible....but just wanted to confirm....

Thanks Again