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Find the equations of two straight line passing through the point (0,1) on which the perpendiculars dropped from the point (2,2) are each of unit length.

Find the equations of two straight line passing through the point (0,1) on which the perpendiculars dropped from the point (2,2) are each of unit length.

Grade:11

1 Answers

Arun
25750 Points
3 years ago
Let the equation of straight line be y=mx+c
 
It passes through (0,a)
 
a=m(0)+c
⇒c=a
y=mx+a........(i)
mx−y+a=0
 
Lenght of perpendicular from (2a,2a) is equal to a
 
1+m 
2
 
​ 
 
∣2am−2a+a∣
​ 
 =a
∣2am−a∣=a 
1+m 
2
 
​ 
 
4a 
2
 m 
2
 +a 
2
 −4a 
2
 m=a 
2
 +a 
2
 m 
2
 
3a 
2
 m 
2
 −4a 
2
 m=0
3m 
2
 −4m=0
m(3m−4)=0
⇒m=0, 
3
4
​ 
 
 
Substituting m in (i)
 
m=0
 
y=0(x)+a
y=a
 
m= 
3
4
​ 
 
 
y= 
3
4
​ 
 x+a
3y=4x+3a....(ii)
 
Slope of line perpendicular to (i) is
 
m 
′
 =− 
4
3
​ 
 
 
Equation of line passing (2a,2a) and slope m 
′
  is
 
y−2a=− 
4
3
​ 
 (x−2a)
4y−8a=−3x+6a
3x+4y=14a......(iii)
 
Feet of perpendicular is point of intersection of (ii) and (iii)
 
Solving (ii) and (iii), we get
 
P( 
5
6a
​ 
 , 
5
13a
​ 
 )
 
Feet of perpendicular from (2a,2a) on y=a is
 
Q(2a,a)
 
Equation of PQ is 
 
y−a= 
5
6a
​ 
 −2a
5
13a
​ 
 −a
​ 
 (x−2a)
y−a=−2(x−2a)
y+2x=5a
 
Hence proved.

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