# Equations of straight lines

(2+5k)x-(3+4k)y-7k=0...(*) is a familt of st. line passing through a fixed point P, where k is a real number

a.Find the coordinates of P

b.A straight line passes through P but it can't be represented by(*) for any real number k. Write down the equation of this st. line

c.Find the equation of the st. line in the family of(*)with slope3/4

Another family of st. lines is given by equation 4x+3y+c=0...(**) where c is a real number.

d.What is the relation of the st. lines in the family of (**)

e.The area of a striangle formed by a-axis, the line in(c) and the one line in

the family of(**) is 4/3sq units.Find the two possible equation of the lines in the family of(**)

f.What kinds of triangles are formed in(e)

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a. (*) can be written as 2x-3y+k(5x-4y-7)=0

∴2x-3y=0------(1)

5x-4y-7=0--------(2)

(1)*5/2

5x=15/2y------(3)

Sub(3)in (2)

7/2y=7

y=2

x=3

∴P=(3,2)

b.The equation of the st. line=5x-4y-7=0

c.Slope of (*)=(2+5k)/(3+4k)

∴(2+5k)/(3+4k)=3/4

8+20k=9+12k

k=1/8

∴The required equation is (21/8)x-(7/2)y-7/8=0

i.e. 21x-28y-7=0

d. The lines in (**) are parallel to each other

e.Slope of line in c=3/4, and slope of line for (**)=-4/3

∴They are perpenducular to each other.

Let the intercept point of the two line are (a,b), where (a,b) satisify both equation

Sub y=0 on 21x-28y+7=0

x=1/3

Sub y=0 in 4x+3y+c=0

x=-c/4

∴sqrt[(1/3-a)²+(0-b)²]*sqrt[(-c/4-a)²+(0-b)²]=8/3

[(1/3-a)²+b²][(-c/4-a)²+b²]=8/3

f. Right-angled triangle.

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• c (2+5k)/(3+4k)=3/4

8+2k=9+12k

10k=-1

k=-1/10

EQ of * 1.5x-2.6y+7/10=0

15x-26y+7=0

d they are all parallel

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