Question

Prove that the eqations represent two straight lines through theorigin (b) 7x2 + 10xy - 8y2 = 0​

Answer 1

Answer:

The general eqaution of second degree ax

2

+2hxy+by

2

+2gx+2fy+c=0

Represents a pair of straight lines if Δ=abc+2fgh−af

2

−bg

2

−ch

2

=0

For 3y

2

−8xy−3x

2

−29x+3y−18=0

a=−3,b=3,h=−4,g=

2

−29

,f=

2

3

,c=−18

Δ=(−3×3×−18)+(2×

2

3

×

2

−29

×−4)−(−3×

2

3

×

2

3

)−(3×

2

−29

×

2

−29

)−(−18×−4×−4)

Δ=162+174+

4

27

−

4

2523

+288

Δ=624−624=0

∴ the equation represents a pair of straight lines.

The point of intersection is found by partially differentiating the equation first with respect to x and then with respect to y and solving both the equations.

∂x

∂

(3y

2

−8xy−3x

2

−29x+3y−18=0)

0−8y−6x−29+0−0=0

6x+8y+29=0....(i)

∂y

∂

(3y

2

−8xy−3x

2

−29x+3y−18=0)

6y−8x−0−0+3−0=0

8x−6y−3=0 .... (ii)

Solving (i) and (ii)

we get x=

2

−3

,y=

2

−5

∴ The point of intersection is (

2

−3

,

2

−5

)

Angle between a pair of straight lines that is tanθ=

∣

∣

∣

∣

∣

∣

a+b

2

h

2

−ab

∣

∣

∣

∣

∣

∣

tanθ=

∣

∣

∣

∣

∣

∣

∣

−3+3

2

(−4)

2

−(−3)(3)

∣

∣

∣

∣

∣

∣

∣

=

0

2(5)

=∞