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Q)

(i) Show that the second degree equations x2−5xy+4y2+x+2y−2=0 represent

(i) Show that the second degree equations x2−5xy+4y2+x+2y−2=0 represents a pair of straight lines.(ii) Find the equations of the individual lines and their point of intersection. Comment
A)

We have x2 – 5xy + 4y2 + x + 2y – 2 = 0 ———————- (1)

Comparing it with  ax2 + 2hxy + by2 + 2gx + 2fy +c = 0 ,

We get,  a = 1 , b = 4, c = – 2, h = – 5/2, g = 1/2 , f = 1.

And  abc + 2fgh – af2 – bg2 – ch2 = –8 + 2 × 1 × 1/2 × (–5/2) –1(1)2 –4(1/2)2 + 2(–5/2)2

= – 8 – 5/2 – 1 – 1 + 25/2 = 0

Hence the given equation represents a pair of straight lines.      [Proved.]

Writing (1) as a function of x , we get

x2 + x(–5 y + 1) + (4y2 + 2y – 2) = 0

Or,                   x = [– (–5y + 1 ) ± √{(–5y + 1)2 – 4 (4y2 + 2y – 2 )}]/2

= [5y – 1 ± √{25y2 + 1 – 10y – 16 y2 – 8y +8}]/2

= [5y – 1 ± √{9y2 – 18y + 9}]/2

= [5y – 1 ± 3 (y – 1)]/2

Or,          2x = 5y – 1 ± (3y – 3)

Or,          2x – 8y + 4 = 0     &       2x – 2y – 2 = 0