Points (a, b) and (c, d) lie on line L in the coordinate plane. Does p

VERITAS PREP OFFICIAL SOLUTION

Correct Answer: C

Statement 1 tells us that the line passes through (0, 0), but does not tell us whether it passes through (3, 3). Accordingly, this statement is insufficient.

Multiplying each side of this equation in statement 2 by -1, we get d - b = c - a. This means that the slope of the line, (d−b)/(c−a) = 1.

(Remember: the slope is equal to the difference in the y-coordinates divided by the difference in the x-coordinates.) This does not, however, tell us whether the line passes through (3, 3), because we don't have the coordinates of any point on the line. This statement is, therefore, insufficient

When taken together, the statements indicate that line L passes through (0, 0) and has a slope of 1. Knowing a point on the line as well as the slope will enable us to plot the entire line, and determine whether it passes through (3, 3). Therefore, the statements together are sufficient, and the correct answer is C.

Points (a, b) and (c, d) lie on line L in the coordinate plane. Does point (3, 3) also lie on line L?

(1) Line L passes through the point of origin.
(2) b - d = a - c.


Answer is C ; Not sure if i am correct in my approach here ;

Stmt 1 : says line passes thru origin so equation of the line can be y = x and y = -x ; Not Suff

stmt 2. b - d = a - c
=> we dont know what values are ;

combining ;

if b - d = a - c
=> b+c =a+d;
there is some scenarios ; either (a,b) be 2,3 and (c,d) be 3,2 which cant be the case as eqn can be y = x or y= -x

now if a point is on a line it must satisfy the eqn ; so the plugin values so should satisfy y = x or y = -x ;

also if we assume (a,b) is lets say if the values were :- (a,b) be (-3,3 ) and (c,d) be (-6, 6 ) which does satify the eqn y = -x but it fails to satisfy b+c = a+d ;

so the only possibility is for the eqn = > y = x ; and the values 3,3 does satisfy hence C ;


I am sorry if my explanation is weird here ; Coordinate is somewhat my weak area

I am not sure if I got this one , can anyone help me to better understand the question ?

Statement B tells us that the line has slope 1, a line with a slope of 1 makes an angle of 45 degrees with the x axis. Hence this is the line y=x . Now obviously point 3, 3 will lie in the line y=x, hence I thought B is sufficient on its own.

Now the argument one can make is we don't know where the line begins and where it ends . It could begin at -3, -3 and end at the origin hence the answer would be no, or the line could end at 4,4 then the answer would be yes.

But even after taking both together how do we know that the line passes through 3,3 or not .

line could begin at -3,-3 and end at 1,1 satisfies both the statement and answer is no.
line could begin at -3,-3 and end at 4,4 satisfies both the statement and answer is yes.

so the answer looks like E, is there anything wrong I am doing here?

Hi stne,

What is a straight line or line?
Definition:- A geometrical object that is straight, infinitely long and infinitely thin. Its location is defined by two or more points on the line whose coordinates are known.

What is a line segment?
Definition:- A line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints.

Here the question stem and its statement talks about a straight line not about a line segment.



if we draw graph in accordance with your statement we would end up with a line segment rather than a straight line. (because you have mentioned a start and an end point of a line)(By definition, we can't restrict the end points of a straight line)
Moreover, if we extend the above segment to infinity at both of its ends, then it would yield a straight line passing through (-4,-4),(-3,-3),(0,0), and (3,3) since the slope(=rise/run) at any of these points is 1.
So, the line L joining the points(a,b) and (c,d), satisfying the conditions of st1&2, passes through the point(3,3).
Graph is enclosed for your easy reference.

Ans. (C)

Hope it helps.

I got most of what you are trying to say, but why is B insufficient ? A line having a slope 1 is the line y =x , hence point (3,3) will lie on this line . So why is the answer not B , as you have already said , a line has infinite length , so it should pass through (3,3) shouldn't it ?

Hi stne,
Passing through origin is the deciding factor here.

St2:- b - d = a - c.
or, d-b=c-a
or, \(\frac{(d-b)}{(c-a)}=1\)
Or, Rise=Run.
We have no reference point here, we can say, rise=run=n (where 'n' is any number)
a) (a,b) & (c,d): (4,5) & (6,7)
b) (a,b) & (c,d): (7,5) & (6,4)
There are infinite no of lines that don't pass through (3,3)
c) (a,b) & (c,d): (0,0) & (6,6)
d) (a,b) & (c,d): (0,0) & (3,3)
At the above condition(when one of the points is origin), the line L passes through point(3,3).

So, the question stem is inconsistent.

hence option(B) is insufficient.

P.S:-
1) Slope form of a straight line in xy-plane: y=mx+c
2) when slope, m=1, then the equation of line: y=x+c (NOT y=x), Therefore, in st2, we can have infinite no of straight lines that doesn't pass through (3,3)
3) When the line passes through origin(with slope=1), y-intercept,c=0; so equation of line: y=x+0=x or y=x

Great starting to make sense now , so one reconfirmation,a line having a slope of 1 will not necessarily make an angle of 45 degrees with the x axis , but a line making angle of 45 degrees with the x axis( positive side of x axis )will definitely will have a slope of 1 , is this correct ?

Post sub explanation of my last post:-
1) Slope form of a straight line in xy-plane: y=mx+c
2) when slope, m=1, then the equation of line: y=x+c (NOT y=x), Therefore, in st2, we can have infinite no of straight lines that doesn't pass through (3,3)
3) When the line passes through origin(with slope=1), y-intercept,c=0; so equation of line: y=x+0=x or y=x

Just go through above points.

1) Angle 45 degree with x-axis mean that abscissa(x-position) and ordinate(y-position) are equidistant from the origin. When they are equidistant , rise=run.hence slope=1.

2) Again, Slope=1 means x-position=y-position. So, slope=1=Rise(vertical change)/Run(horizontal change)=Tan(45). If you say, angle made by line & +ve axis, then it may or mayn't be 45 degree but with x-axis, it definitely makes an angle of 45. N.B:- All straight lines(except slope=0 and undefined ) cut both x-axis & y-axis, so, the line with slope 1 will cut x-axis(+ve or -ve or at origin) making an angle of 45 with it.

stne, please go through. Why did you say "a line having a slope of 1 will not necessarily make an angle of 45 degrees with the x axis". Do you have any points?

Right , so a line having a slope 1 will always make an angle of 45 degrees with the x axis , but at which part can only be defined, if a point is given.
So as you have very nicely shown in the diagram, slope of 1 will make an angle of 45 degree with the x axis but it may not necessarily be the line y=x.
The line will be y=x only when slope is 1 and it passes through the origin or any point such as(-3,-3).. (1,1) (2,2) ..(6,6)etc . Hope now I have arrived at the correct conclusion.

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