IIT JEE , AIEEE Forum - Mathematics , Analytical Geometry

danny_007

Find the equation of straight lines passing through the point (2,3) and having an intercept of length 2 unites between the straight lines 2x + y = 3 , 2x + y = 5

i knw a long method for this can u provide any shorter method?

dheer_07

ans is x-2y=-4

plot the lines on coord systemm n ull get ur result

plz rate if im crrect!!

cheers

akhil_o

the two given lines are parallel, with perpendicular distance equal to
2/root(5)
if we draw the triangle between the point of contacts with 2 lines and ,
we get the third side=4/root 5
and tan A=1/2 and for second line -1/2

so simply find the equaation of lines passing through (2,3) and making angle tan inverse 1/2 and -1/2
with m=-2

dheer_07

hey akhil i think slope will be 1/2 not 2!!!

akhil_o

i have written 1/2 ...
that the slope wrt y+2x is 1/2 and -1/2

if slope is tan X

tan X=(2+1/2)/(1-2*1/2)
=5/2

and tan X=(2-1/2)/(1+2*1/2)
=3/4

so the equations should be

y=5x/2+c and y=3x/4+d
such that they pass thru (2,3)

y=5x/2-2 and y=3x/4+3/2

ananth_patri

@akhil_o i am at loss to understand u r procedure plz explain clearly.....how did u get the triangle with two parallel line????

May be we can do this prob ....assuming the equn be y=mx+c and the intercept betw the lines is by finding the point of intersect of lines and req line and solving it.....but its a lengthy process....

akhil_o

does this make it clear?

ananth_patri

ohh!!! u did a gr8 deal of thinkin on it???but how a non routine way of solvin it,,,,,,no one could think it in one go(atleast I)....nice work buddy

chimanshu_007

any line through (2,3) is given by

x - 2 / cos@ = y - 3 / sin@ = r

let this line cuts 2x + y = 5 and 2x + y = 3 at D and C resp. (look in the figure)

we are given DC = 2

D = 2 + r cos@ , 3 + r sin@

C = ( 2 + (r + 2)cos@ , 3 + (r + 2)sin@)

As D , C lie on 2x + y = 5 , 2x + y = 3

2(2 + rcos@) + (3 + rsin@) = 5

2 ( 2 + (r + 2) cos@) + (3 + (r + 2) sin@) = 3

Subtract the above 2 =ns

4 cos@ + 2 sin@ =- 2

2cos@ + sin@ = -1

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