+91-705-589-3577
FlagBACK
Flag Analytical Geometry> a straight line L with negative slope pas...
question mark

a straight line L with negative slope passes through the point (8,2) and cuts positive coordinate axes at the points P and Q. Find the minimum value of OP + OQ as L varies where .

Mohammed Junaid Mohiuddin , 9 Years ago
Grade 11
anser 2 Answers
Vikas TU
a straight line L with negative slope having let intercepts with x  and y respectively be:
a and b.
Therefore, eqn. of the line passing through (8,2) is:
8/a + 2/b = 1
or
a = 8b/b-2 …...................(1)
 
minimum value of OP + OQ
is: a + b = M (let)
from  eqn. (1)
M = b + 8b/b-2
dM/db = 1 + 8{(b-2) – (b)}/(b-2)^2 = 0
b^2 + 3b – 12 = 0
(b-3) (b + 4) = 0
b =3 since it is on positive y axis.
put y in eqn. (1)
u will get, 
M = 27 minimum value. hennce!
Last Activity: 9 Years ago
Bob
from point slope form we have:
 let the slope of ine L be -m.
y1 = 2, x1 = 8.
 y-2 = -m(x-8)
 
y -2 = -mx + 8m
mx + y = 8m + 2
 
mx/(8m+2)/m + y/(8m+2) = 1
 
therefore, OP = (8m + 2)/m    OQ = 8m + 2
 
Let S = OP + OQ
 
(8m + 2)/m + 8m + 2 
 
8 + 2/m + 2 + 8m
 
10 + 2/m + 8/m
 
using completing the square method we get:
 
18 + [ (23/2m1/2 – (2/m)1/2  ]2
 
For minimum value of S 
  [ (23/2m1/2 – (2/m)1/2  ]2   =  0
 
solving we get m= ½
and m= -1/2
 
 
So, minimum value of OP + OQ will be 18 (for m=-1/2 i.e negative slope)
 
 
Last Activity: 8 Years ago
star
LIVE ONLINE CLASSES

Prepraring for the competition made easy just by live online class.

tv

Full Live Access

material

Study Material

removal

Live Doubts Solving

assignment

Daily Class Assignments


Other Related Questions on analytical geometry

question mark
  • The locus of the point represented by x= t^2 +t+1 and y=t^-t+1 is 
analytical geometry
1 Answer Available

Last Activity: 3 Years ago

question mark
show that the centroids of the triangles of which three perpendiculars lie along the lines y=m1x,y=m2x,y=m3x lie on y(3+m2m3+m3m2+m1m2)=x(m1+m2+m3+3m1m2m3)
analytical geometry
1 Answer Available

Last Activity: 3 Years ago

question mark
show that the equation (ab-h²)(ax²+2hxy+by²+2gx+2fy)+af²+bg²-2fgh=0 represent a pair of straight lines and that these straight lines form a rhombus with the lines ax²+2hxy+by²=0
analytical geometry
1 Answer Available

Last Activity: 3 Years ago

question mark
A circle has its center on straight line 5x-2y+1=0 and cuts the x axis at two points whose absciasse are -5,3 find equation of the circle and it's radius. 
 
analytical geometry
1 Answer Available

Last Activity: 3 Years ago

question mark
Prove that ax^2+by^2+cz^2 +2ux+2vy+2wz+d=0 represent a cone if u^2/a+v^/b+w^2/c=d
analytical geometry
1 Answer Available

Last Activity: 3 Years ago