# Pair of Straight Lines

## Short Questions

If the lines represented by the equation 16x2 - kxy + 9y2 = 0 are coincident, find the value of k.

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.

If the lines represented by the equation kx2 - 24xy + 16y2 = 0 are coincident, find the value of k.

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.

If the lines represented by the equation 2x2 + 8xy- (m + 1)y2 = 0 are coincident, find the value of 'm'.

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.

If the pair of lines represented by the equation x2 + 2kxy + 4y2 = 0 are coincident, find the value of k.

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.

Find the value of 'k' if the pair of lines represented by 3x2 - 6xy + (k + 4)y2 = 0 are coincident to each other.

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.

Find the single equation which represents a pair of straight lines x = 3y and 3x = y.

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.

Find the separate equations of the pair of lines represented by equation x2 + 2x + 2y - y2 = 0.

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.

Prove that the angle between a pair of lines represented by 5x2 - 6xy - 5y2 = 0 is a right angle.

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.

Prove that the pair of lines represented by the equation 16x2 - 24xy + 9y2 = 0 are coincident.

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.

Find the separate equations of the pair lines represented by the equation x2 - 2x - y2 + 2y = 0.

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.

Find the value of m, if the lines represented by equation (m - 6)x2 - 8xy + my2 = 0 are coincident.

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.

If a pair of straight lines represented by the equation kx2 - 12xy + 3y2 = 0 are coincident, find the value of k.

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.

If the pair of lines represented by an equation mx2 - 5xy - 6y2 = 0 are perpendicular to each other, find the value of m.

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.

Find the angle between the pair of lines represented by equation, 3x2 + 7xy + 2y2 = 0.

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°.