Find the point which lies on the perpendicular bisector of the line segment joining the points A (1, 5) and
B (4, 6) cuts the y-axis.
Answers
Answer:
0,13)
(0,-13)
(0,12)
(13,0)
Answer :
a
Solution :
Firstly, we plot the points of the segment on the paper and join them.
We know that , the Perpendicular bisector of the line segment AB bisect the segment AB, i.e., perpendicular bisector of line segment AB passes through the mid - point of AB.
∴ Mid - [point of AB (1+42,5+62)
⇒P=(52,112)
[∴ mid point of line segment passes through the points (x1+y1) and (x2+y2)=(x1+x22,y1+y22)]
Now , we draw a straigth line on paper passes through the mid - point P . We see that the perpendicular bisector cuts the Y- axis at the point (0,13). Hence, the required point is (0,13).
Step-by-step explanation:
Alternate Method
We know that , the equation of line which passes through the points (x1,y1)and(x2,y2) is
(y−y1)=y2−y1x2−x1(x−x1) ...(i)
Here, x1=,y1=5andx2=4,y2=6
So , the equation of line segment joining the points A(1,5) and B (4,6) is
(y−5)=6−54−1(x−1)
⇒(y−5)=13(x−1)
⇒3y−15=x−1
⇒3y=x−14⇒y=13x−143 ...(ii)
∴ Slope of line segment , m1=13
If two lines are perpendicular to each other , then the relation between its slopes is
m1⋅m2=−1 ...(iii)
where, m1 = Slope of line 1
and = Slope of line 2
Also, we know that the perpendicular bisector of the line segment is perpendicular on the line segment. Let slope of line segment is m2.
From Eq. (iii),
m1⋅m2=13⋅m2=−1
⇒m2=−3
Also we know that perpendicular bisector is passes through the mid- point of line segment.
∴ Mid - point of line segment =(1+42,5+62)=(52,112)
Equation of perpendicular bisector , which has slope (-3) and passes through the point (52,112), is
(y−112)=−(−3)(x−52)
[Since , equation of line passes through the point (x1,y1) and having slope m (y−y1)=m(x−x1)]
⇒(2y−11=−2(2x−5)
⇒2y−11=−6x+15
⇒6x+2y=26
⇒3x+y=13 ...(iv)
If perpendicular bisector cuts the y- axis , then put x=0 in Eq. (iv),
3×0+y=13⇒y=13
So , the required point is (0,13).
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