6x2-15y2-xy+16x+24y=0

Question

represent 2 srtaight lines find the equation of each straight lines???

 

Answer 1

6x^2-15y^2-xy+16x+24y=0 represents 2 straight lines. Find the equation of each straight line.

Let x = 0:

6(0)^2-15y^2-(0)y+16(0)+24y=0

-15y^2+24y=0

y(5y-8)=0

y = 0 or y = 8/5

(0,0) is on y = mx+0 and
(0,8/5) is on x/a + y/(8/5) = 1.

Let y = 0:

6x^2-15(0)^2-x(0)+16x+24(0)=0

6x^2+16x=0

x(3x+8)=0

x = 0 or x = –8/3

(0,0) is on y = mx+0 and
(–8/3,0) is on x/(–8/3) + y/(8/5) = 1

–3x/8 + 5y/8 = 1

3x – 5y = –8 is one line

Let x = 1:
y = mx ==> y = m(1) = m

6(1)^2-15(m)^2-(1)(m)+16(1)+24(m)=0

6-15m^2-m+16+24m=0

-15m^2+22+23m=0

15m^2–23m–22=0

m = (23 ± √(23^2–4(15)(–22)))/(2(15))

m = (23 ± √(529+1320))/30

m = (23 ± √(1849))/30

m = (23 ± 43)/30 = –20/30 or 66/30

m = –2/3 or 11/5

y = –2/3 x or y = 11/5 x

2x + 3y = 0 or 11x – 5y = 0 is second line

3x – 5y + 8 = 0 is first line

(2x + 3y)(3x – 5y + 8) = (0)(0)

2x(3x – 5y + 8) + 3y(3x – 5y + 8) = 0

6x^2 – 10xy + 16x + 9xy – 15y^2 + 24y = 0

6x^2 – 15y^2 – xy + 16x + 24y = 0
Compare to original equation:
6x^2 - 15y^2 - xy + 16x + 24y = 0 √

So equations of the two lines are:

2x + 3y = 0 and 3x – 5y = –8