Pair of Straight Lines

Here,
Given equation: 16x² - kxy + 9y² = 0 --- (i)

Comparing it with ax² + 2hxy + by² = 0, we get,
a = 16, b = 9, h = $\frac{-k}{2}$

Since, lines represented by (i) are coincident, h² = ab or, $(\frac{-k}{2})^2$ = 16 × 9
or, k² = 144 × 4
or, k = $\sqrt{576}$
so, k = ±24

Hence, value of k is ±24.

Here,
Given equation: kx² - 24xy + 16y² = 0 --- (i)

Comparing it with ax² + 2hxy + by² = 0, we get,
a = k, b = 16, h = -12

Since, lines represented by (i) are coincident, h² = ab
or, (-12)² = k × 16
or, $\frac{144}{16}$ = k
so, k = 9

Hence, value of k is 9.

Here,
Given equation: 2x² + 8xy- (m + 1)y² = 0 --- (i)

Comparing it with ax² + 2hxy + by² = 0, we get,
a = 2, b = -(m + 1), h = 4

Since, lines represented by (i) are coincident, h² = ab
or, 4² = -2(m + 1)
or, 16 = -2m - 2
or, 18 = -2m
so, m = -9

Hence, value of 'm' is -9.

Here,
Given equation: x² + 2kxy + 4y² = 0 --- (i)

Comparing it with ax² + 2hxy + by² = 0, we get,
a = 1, b = 4, h = k

Since, lines represented by (i) are coincident, h² = ab
or, k² = 1 × 4
or, k = $\sqrt4$
so, k = ±2

Hence, value of 'm' is ±2.

Here,
Given equation: 2x² + 8xy- (m + 1)y² = 0 --- (i)

Comparing it with ax² + 2hxy + by² = 0, we get,
a = 3, b = k + 4, h = -3

Since, lines represented by (i) are coincident, h² = ab
or, (-3)² =3 (k + 4)
or, 9 = 3k + 12
so, k = -1

Hence, value of 'k' is -1.

Here,
Given equations:
x - 3y = 0 --- (i)
3x - y = 0 --- (ii)

Now, multiplying (i) and (ii), we get, (x - 3y)(3x - y) = 0 or, 3x² - xy - 9xy + 3y² = 0
so, 3x² - 10xy + 3y² = 0

Hence, 3x² - 10xy + 3y² = 0 is the required single equation.

Here, Given equation,

x² + 2x + 2y - y² = 0
or, x² + 2x + 2y - y² = 0
or, x² - y² + 2(x + y) = 0
or, (x + y)(x - y) + 2(x + y) = 0
or, (x + y)(x - y + 2) = 0
Either, x + y = 0 --- (i)
OR, x - y + 2 = 0 --- (ii)

Hence, equation (i) and (ii) are separate equation of pair of lines represented by given equation.

Here,
Given equation: 5x² - 6xy - 5y² = 0 --- (i)

Comparing it with ax² + 2hxy + by² = 0, we get,
a = 5, b = -5, h = -3

Now, θ be the angle between lines represented by equation (i), so, tanθ = $\frac{\pm 2\sqrt{h^2\text{ - ab}}}{\text{a + b}}$ or, tanθ = $\frac{\pm 2\sqrt{h^2\text{ - ab}}}{\text{5 - 5}}$
or, tanθ = tan90°
so, θ = 90°

Hence, angle between a pair of lines represented by given equation is a right angle.

Here,
Given equation: 16x² - 24xy + 9y² = 0 --- (i)

Comparing it with ax² + 2hxy + by² = 0, we get,
a = 16, h = -12, b = 9

Now, for coincident condition, h² = ab
or, (-12)² = 16 × 9
so, 144 = 144

Hence, pair of lines represented by given equation is coincident.

Here, Given equation,
x² - 2x - y² + 2y = 0 or, x² - y² - 2x + 2y = 0
or, (x + y)(x - y) - 2(x - y) = 0
or, (x - y)(x + y - 2) = 0
Either, x - y = 0 --- (i)
OR, x + y - 2 = 0 --- (ii)

Hence, equation (i) and (ii) are separate equation of pair of lines represented by given equation.

Here,
Given equation: (m - 6)x² - 8xy + my² = 0 --- (i)

Comparing it with ax² + 2hxy + by² = 0, we get,
a = m - 6, h = -4, b = m

Since, lines represented by (i) are coincident, h² = ab
or, (-4)² = (m - 6)m
or, 16 = m² - 6m
or, m² - 6m - 16 = 0
or, m² - (8 - 2)m - 16 = 0
or, m² - 8m + 2m - 16 = 0
or, m(m - 8) + 2 (m - 8) = 0
or, (m - 8)(m + 2) = 0
Either, m = 8
OR, m = -2

Hence, value of 'm' is 8 or -2.

Here,
Given equation: kx² - 12xy + 3y² = 0 --- (i)

Comparing it with ax² + 2hxy + by² = 0, we get,
a = k, h = -6, b = 3

Since, lines represented by (i) are coincident, h² = ab
or, (-6)² = k × 3
so, k = 12

Hence, value of 'k' is 12.

Here,
Given equation: mx² - 5xy - 6y² = 0 --- (i)

Comparing it with ax² + 2hxy + by² = 0, we get,
a = m, h = -$\frac 52$, b = -6

Since, lines represented by (i) are perpendicular, a + b = 0
or, m - 6 = 0
so, m = 6

Hence, value of 'm' is 6.

Here,
Given equation: 3x² + 7xy + 2y² = 0 --- (i)

Comparing it with ax² + 2hxy + by² = 0, we get,
a = 3, h = $\frac 72$, b = 2

Now, θ be the angle between lines represented by equation (i), so, tanθ = $\frac{\pm 2\sqrt{h^2\text{ - ab}}}{\text{a + b}}$ or, tanθ = $\frac{\pm 2\sqrt{(\frac72)^2\text{ - 6}}}{\text{3 + 2}}$
or, tanθ = $\frac{\pm 2\sqrt{49\text{ - 24}}}{2 \times 5}$
or, tanθ = ±$\frac55$
or, tanθ = ±1

Taking positive,
or, tanθ = tan45°
so, θ = 45°

Taking negative,
or, tanθ = tan135°
so, θ = 135°

Hence, angle between a pair of lines represented by given equation is 45° or 135°.