Question Statement Find the measure of the acute angle between the pair of lines represented by the following homogeneous equations:
(i) x 2 − y 2 = 0
(ii) x 2 + 5 x y + 4 y 2 = 0
(iii) 15 x 2 − 19 x y + 6 y 2 = 0
(iv) 10 x 2 − x y − 9 y 2 = 0
(v) 5 x 2 − 3 x y − 2 y 2 = 0
(vi) 7 x 2 + 2 x y − 9 y 2 = 0
Background and Explanation When a second-degree homogeneous equation a x 2 + 2 h x y + b y 2 = 0 θ tan θ = ∣ a + b ∣ 2 h 2 − ab a + b = 0 θ = 90° tan θ
Solution First, we compare the given equation with the standard form a x 2 + 2 h x y + b y 2 = 0
a = 1 , 2 h = 0 ⇒ h = 0 , b = − 1
Using the formula for the angle between the lines:
tan θ = a + b 2 h 2 − ab
Substituting the values:
tan θ = 1 + ( − 1 ) 2 0 2 − ( 1 ) ( − 1 ) = 0 2 1 = 0 2 = ∞
Since tan θ = ∞
θ = tan − 1 ( ∞ ) = 9 0 ∘
Interpretation: When the denominator a + b = 0
Comparing with the standard form a x 2 + 2 h x y + b y 2 = 0
a = 1 , b = 4 , 2 h = 5 ⇒ h = 2 5
Applying the angle formula:
tan θ = a + b 2 h 2 − ab = 1 + 4 2 ( 2 5 ) 2 − ( 1 ) ( 4 )
Simplifying the expression under the square root:
= 5 2 4 25 − 4 = 5 2 4 25 − 16 = 5 2 4 9
= 5 2 ( 2 3 ) = 5 3
Therefore:
θ = tan − 1 ( 5 3 ) ≈ 30.9 6 ∘ ≈ 3 1 ∘
Comparing with the standard form:
a = 15 , b = 6 , 2 h = − 19 ⇒ h = − 2 19
Using the angle formula:
tan θ = a + b 2 h 2 − ab = 15 + 6 2 ( − 2 19 ) 2 − ( 15 ) ( 6 )
Calculating step by step:
= 21 2 4 361 − 90 = 21 2 4 361 − 360 = 21 2 4 1
= 21 2 ( 2 1 ) = 21 1
Thus:
θ = tan − 1 ( 21 1 ) ≈ 2.7 3 ∘
Comparing with the standard form:
a = 10 , b = − 9 , 2 h = − 1 ⇒ h = − 2 1
Applying the formula:
tan θ = a + b 2 h 2 − ab = 10 + ( − 9 ) 2 ( − 2 1 ) 2 − ( 10 ) ( − 9 )
Simplifying:
= 1 2 4 1 + 90 = 2 4 1 + 360 = 2 4 361
= 2 ( 2 19 ) = 19
Therefore:
θ = tan − 1 ( 19 ) ≈ 86.9 8 ∘ ≈ 8 7 ∘
Comparing with the standard form:
a = 5 , b = − 2 , 2 h = − 3 ⇒ h = − 2 3
Using the angle formula:
tan θ = a + b 2 h 2 − ab = 5 + ( − 2 ) 2 ( − 2 3 ) 2 − ( 5 ) ( − 2 )
Calculating the values:
= 3 2 4 9 + 10 = 3 2 4 9 + 40 = 3 2 4 49
= 3 2 ( 2 7 ) = 3 7
Thus:
θ = tan − 1 ( 3 7 ) ≈ 66.8 0 ∘
Comparing with the standard form:
a = 7 , b = − 9 , 2 h = 2 ⇒ h = 1
Applying the angle formula:
tan θ = a + b 2 h 2 − ab = 7 + ( − 9 ) 2 ( 1 ) 2 − ( 7 ) ( − 9 )
Simplifying:
= − 2 2 1 + 63 = − 2 2 64 = − 2 2 ( 8 ) = − 8
Since we require the acute angle between the lines, and the tangent of an acute angle must be positive, we take the absolute value:
tan θ = 8
Therefore:
θ = tan − 1 ( 8 ) ≈ 82.8 7 ∘
Standard Form: a x 2 + 2 h x y + b y 2 = 0 Angle Formula: tan θ = a + b 2 h 2 − ab Special Case: If a + b = 0 tan θ = ∞ θ = 9 0 ∘ Sign Convention: If calculation yields negative tan θ ∣ tan θ ∣
Summary of Steps
Identify coefficients: Compare the given equation with a x 2 + 2 h x y + b y 2 = 0 a h b Check for perpendicularity: If a + b = 0 θ = 9 0 ∘ Apply formula: Calculate tan θ = a + b 2 h 2 − ab Simplify: Evaluate the discriminant h 2 − ab Determine sign: If tan θ < 0 Calculate angle: Compute θ = tan − 1 ( value )