A line perpendicular to the line segment joining the points A ( 1 , 0 ) and B ( 2 , 3 ) , divides it at C in the ratio of 1 : 3 . Then the equation of the line is
Answers
According to the section formula, the coordinates of the points that divides the line segment joining the points (1,0) and (2,3) in the ratio 1:n is given by {
1+n
n(1)+1(2)
,
1+n
n(0)+1(3)
}={
n+1
n+2
,
n+1
3
}
The slope of the line joining the points (1,0) and (2,3) is m=
2−1
3−0
=3.
We know that two non-vertical lines are perpendicular to each other if and only if their slpoes are negative reciprocals of each other.
Therefore, slope of the line that is perpendicular to the line joining the points (1,0) and (2,3) is =−
m
1
=−
3
1
Now the equation of the line passing through ={
n+1
n+2
,
n+1
3
} and whose slope is −
3
1
is given by, {y−
n+1
3
}=
3
−1
{x−
n+1
n+2
}
⇒3[(n+1)y−3]=−[x(n+1)−(n+2)]
⇒3(n+1)y−9=−(n+1)x+n+2
⇒(1+n)x+3(1+n)y=n+11
Step-by-step explanation:
The given lines are perpendicular and as AB = AC , Therefore △ ABC is art . angled isosceles . Hence the line BC through ( 1 , 2) will make an angles of ±45
∘
with the given lines . Its equations is y - 2 = m (x - 1) where m = 1 / 7 and -7 as in .Hence the possible equations are 7x + y - 9 = 0 and x - 7y + 13 = 0
Alt :
The two lines will be parallel to bisectors of angle between given lines and they pass through ( 1, 2)
∴ y - 2 = m ( x - 1)
where m is slope of any of bisectors given by
5
3x+4y−5
=±
5
4x−3y−15
or x - 7y + 13 = 0 or 7x + y - 20 = 0
∴ m = 1 / 7 or - 7
putting in (1) , the required lines are 7x + y - 9 = 0
and x - 7y + 13 = 0 as found above