What is the equation that represents the locus of points equidistant

Question

What is the equation that represents the locus of points equidistant from the points A (-2, 1) and B (6, 3)?

(A) 4x+y=20
(B) y+4x=10
(C) 4y+x=10
(D) y+2x=20
(E) x+4y=40

chetan2u · Accepted Answer

hi,

you can work on two issues and get your answer..

1) A (-2,1) and B (6,3).. 
therefore a point equidistant from BOTH will be there average..
so x = \frac{-2+6}{2}= 2  and  y =\frac{1+3}{2} = 2 ..
substitute x as 2 and y as 2 and see which equation satisfies the values...
 ONLY B and C are left..

2) As shown by you, work on slope.. 
slope of line AB = \frac{3-1}{6-(-2)} = \frac{2}{8} = \frac{1}{4} ..
so slope of line joining all points = -4, as this line will be perpendicular to AB...
 ONLY A and B satisfy the condition

ONLY B satisfies both points.. 
ans B

Oski · Answer

Answer is (B)

The locus of points equidistant from 2 points is the perpendicular bisector of the segment between these 2 points.

Vector AB is (8,2) and we want a vector (x,y) which is normal to that one (it will give the direction of the perpendicular bisector).

Their dot product has to be null, so we get 8x+2y=0, which is equal to 4x+y=0

Therefore any line with equation 4x+y=K is perpendicular to the line that goes through A and B.

Answer is then either (A) or (B).

If the line we are looking for is the locus of points equidistant from A and B, it contains the middle point between A and B, which is M(2,2).

This point verifies 4x+y=10: (B) is the answer.

alpha_plus_gamma · Answer

B

^2 +  ^2 =  ^2 +  ^2

the squared terms on both sides get cancelled leaving us with B

jallenmorris · Answer

Maybe I just like to work harder than everyone else. Here is how I arrived at B.

Locus = perpendicular bisector.

Find the mid point between A & B. 6 - -2 = 8 and 3 - 1 = 2, so go right 8 and up 2, so midpoint is half that. Go right 4, and up 1. Brings us to {2, 2}. Slope of AB is 1/4 because rise over run 2 up 8 right =  \frac{2}{8}  or  \frac{1}{4} .

The prependicular line to this will be inverted and the sign flipped. Inverse of 1/4 is 4/1 and negative is -4. To find the x-intercept you have to move left from 2,2 so go up 4, but left, or negative 1. (If you have 4/1, if you move in a positive direction for one of the numbers 4 or 1, the other must move in a negative direction). Up 4 and left 1 brings you to point 1, 6, that's not the x yet, so up 4 and left 1 again brings us to 0 ,10. x-intercept of +10.

Now this perpendicular line is represented by the equation y = -4x + 10. Find the answer that will give you that when solved for y. This is B

y + 4x = 10
y = 10 - 4x or
y = -4x +10

ritula · Answer

OA is B indeed. i was doing the big mistake of keeping its slope the same as the given line while actually it is a perpendicular line. Thanks to all!

ruchi857 · Answer

What is the equation that represents the locus of points equidistant from the points A (-2,1) and B (6,3)

A) 4x+y = 20
B) y+4x = 10
C) 4y+x = 10
D) y+2x = 20
E) x+4y = 20

My Solution is to find the slope of line AB, which is equal to 1/4. The equation of locus of points asked thus will have a slope of -4.

That itself allows me to reject C,D and E

Bunuel · Answer

____________________
Merging topics.

MBA20 · Answer

chetan2u  Could you please explain the red part.

cruiseav · Answer

Hi  alpha_plus_gamma

Just curious what logic you have used in the above calculation.
Thanks

Fdambro294 · Answer

Concept: the Equation of the Line that is the PERPENDICULAR BISECTOR of Line Segment AB will contain All the Points that will be EQUI-Distant from Point A and Point B

(1st) Find Slope of Perpendicular Line to Segment AB

Slope of Segment AB = m1 = (1 - 3) / (-2 - 6) = -2 / -8 = +(1/4)

Perpendicular Line through Segment AB will have a (-)Negative Reciprocal Slope:

m1 * m2 = -(1)

+(1/4) * m2 = -(1)

m2 = -4

The Equation of the Perpendicular Line will be of the Form: y = -(4)x + b

(2nd) The Line that is the Perpendicular BISECTOR will pass through the MID-POINT of Segment AB ---- and Every Point on this Perpendicular Bisector Line will be Equi-Distant from Point A and Point B

MID-POINT of Point A and Point B

X-Coordinate: (-2 + 6) / 2 = +2

Y-Coordinate: (1 + 3) / 2 = +2

Thus, the MID-POINT (+2 , +2) will lie on the Perpendicular Bisector Line

(3rd) Find b = Y-Intercept for the Perpendicular Bisector Line

Point (+2 , +2) lies on the Line

the Line Equation is of the Form: y = -(4)x + b

---Plugging in (2 , 2)-----

2 = -4 * 2 + b

2 = -8 + b

b = +10

The Full Equation of the Line that will give all the Points Equi-Distant:

y = -4x + 10

y + 4x = 10

-Answer B-

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What is the equation that represents the locus of points equidistant

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What is the equation that represents the locus of points equidistant

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