? asked in 科學及數學數學 · 1 decade ago

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)

2 Answers

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  • 1 decade ago
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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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  • 1 decade ago

    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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