Distance in Cartesian Plane

[dtweah]

The distance between the point (1,2) and a line in the X-Y plane is (3(2)^0.5)/4. If the y intercept of the line is 5/2 find the slope of the line?

A. -1

B -1/2

C. 1/2

D. 1

E. 2

[Vemuri]

The question stem provides us with 2 points (1,2) & (0,5/2). Using the slope formula ==> ((5/2)-2)/(0-1) ==> -1/2

IMO B

[dtweah]

The question stem provides us with 2 points (1,2) & (0,5/2). Using the slope formula ==> ((5/2)-2)/(0-1) ==> -1/2

IMO B

Give it another shot vemuri.

[Vemuri]

The question stem provides us with 2 points (1,2) & (0,5/2). Using the slope formula ==> ((5/2)-2)/(0-1) ==> -1/2

IMO B

Give it another shot vemuri.

Ok, the slope should be determined by using a point on the line & the y intercept. But, how are we to determine the point? I think I am stuck here.

I am not sure I understand how to use the info that (1,2) is (3(2)^0.5)/4 away from the line on the XY plane.

[dumb.doofus]

Phew!!! Man.. such problems come on GMAT??? I am not sure..
anyways, the solution is in the picture below... login to see..
I think the answer should be 1 i.e. option D

[Vemuri]

Damn !!! Unless there is a easier way to solve this problem, I bet this is not a GMAT question.

[dtweah]

The distance between the point (1,2) and a line in the X-Y plane is (3(2)^0.5)/4. If the y intercept of the line is 5/2 find the slope of the line?

A. -1

B -1/2

C. 1/2

D. 1

E. 2

Altough you could use the distance to a line formula and solve you can also solve without it. Put eqn into y=mx+b y=mx+5/2 or 2y=mx+5. Plug point in (1,2) you can find slope m.
2(2)=m(1)+5
m=1
Choose D.

Using Distance Formula:
Rationalize distance. 3/(2)2^.5
Denominator equal square root of 8=4+4 or 2^2+2^2
So A=+-2 and B=+-2



|Ax +B +C|=|3| From y intercept info you know C=5 and A =2
|2(1)+B(2)+5|=3
|7+2B|=3

The def of absolute value has:
7+2B= 3, for B=-2, this solves.

7+2B=-3, since distance is never negative we ignore this part

The eqn is 2x-2y=5 from which the slope can be determined to be 1.
Choose D.

The first method solved under 1 min. The second about midway over 1.

[avenus]

I believe there's stg wrong with your reasoning:

Altough you could use the distance to a line formula and solve you can also solve without it. Put eqn into y=mx+b y=mx+5/2 or 2y=mx+5. Plug point in (1,2) you can find slope m.
2(2)=m(1)+5
m=1

you're assuming the line will cross (1,2). Why? Distance would then be zero....plus result would be m=-1

The eqn is 2x-2y=5 from which the slope can be determined to be 1.
Choose D.

This eq yields a y-intercept of -5/2, which is different from the 5/2 stated in the problem

[dtweah]

I believe there's stg wrong with your reasoning:

Altough you could use the distance to a line formula and solve you can also solve without it. Put eqn into y=mx+b y=mx+5/2 or 2y=mx+5. Plug point in (1,2) you can find slope m.
2(2)=m(1)+5
m=1

you're assuming the line will cross (1,2). Why? Distance would then be zero....plus result would be m=-1

The eqn is 2x-2y=5 from which the slope can be determined to be 1.
Choose D.

This eq yields a y-intercept of -5/2, which is different from the 5/2 stated in the problem

The latter is a transpositional error. Ax +BY +C= 0 so 2x-2y+5=0. Which has slope 1. The former is a serious error and you are right. (1,2) does not pass through 2x-2y+5=0(1,7/2) does. So the only valid solution is method 2. I spotted the error in V's response when I told him to try again but committed the same error! I think where the error comes from is that in using the formula for finding the distance between a point and a line, we use the point in the equation but this does not mean the point is on the line! Good observation. Thanks.