Prove that the following equation represent two straight lines; find also their point of intersection and the angle between them.
3y2−8xy−3x2−29x+3y−18=0
3y2−8xy−3x2−29x+3y−18=0
The
general eqaution of second degree ax2+2hxy+by2+2gx+2fy+c=0
Represents a pair of straight lines if Δ=abc+2fgh−af2−bg2−ch2=0
For 3y2−8xy−3x2−29x+3y−18=0
a=−3,b=3,h=−4,g=−292,f=32,c=−18Δ=(−3×3×−18)+(2×32×−292×−4)−(−3×32×32)−(3×−292×−292)−(−18×−4×−4)Δ=162+174+274−25234+288Δ=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(3y2−8xy−3x2−29x+3y−18=0)0−8y−6x−29+0−0=06x+8y+29=0....(i)∂∂y(3y2−8xy−3x2−29x+3y−18=0)
6y−8x−0−0+3−0=0
8x−6y−3=0 .... (ii)
Solving (i) and (ii)
we get x=−32,y=−52
∴ The point of intersection is (−32,−52)
Angle
between a pair of straight lines that is tanθ=∣∣
∣∣2√h2−aba+b∣∣
∣∣
tanθ=∣∣
∣
∣∣2√(−4)2−(−3)(3)−3+3∣∣
∣
∣∣=2(5)0=∞⇒θ=π2
The general eqaution of second degree ax2+2hxy+by2+2gx+2fy+c=0
Represents a pair of straight lines if Δ=abc+2fgh−af2−bg2−ch2=0
For 3y2−8xy−3x2−29x+3y−18=0
a=−3,b=3,h=−4,g=−292,f=32,c=−18Δ=(−3×3×−18)+(2×32×−292×−4)−(−3×32×32)−(3×−292×−292)−(−18×−4×−4)Δ=162+174+274−25234+288Δ=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(3y2−8xy−3x2−29x+3y−18=0)0−8y−6x−29+0−0=06x+8y+29=0....(i)∂∂y(3y2−8xy−3x2−29x+3y−18=0)
6y−8x−0−0+3−0=0
8x−6y−3=0 .... (ii)
Solving (i) and (ii)
we get x=−32,y=−52
∴ The point of intersection is (−32,−52)
Angle between a pair of straight lines that is tanθ=∣∣ ∣∣2√h2−aba+b∣∣ ∣∣
tanθ=∣∣ ∣ ∣∣2√(−4)2−(−3)(3)−3+3∣∣ ∣ ∣∣=2(5)0=∞⇒θ=π2
