Advertisements
Advertisements
Question
The value of the λ, if the lines (2x + 3y + 4) + λ (6x – y + 12) = 0 are
| Column C1 | Column C2 |
| (a) Parallel to y-axis is | (i) λ = `-3/4` |
| (b) Perpendicular to 7x + y – 4 = 0 is | (ii) λ = `-1/3` |
| (c) Passes through (1, 2) is | (iii) λ = `-17/41` |
| (d) Parallel to x axis is | λ = 3 |
Solution
| Column C1 | Column C2 |
| (a) Parallel to y-axis is | (i) λ = 3 |
| (b) Perpendicular to 7x + y – 4 = 0 is | (ii) λ = `-17/41` |
| (c) Passes through (1, 2) is | (iii) λ = `-3/4` |
| (d) Parallel to x axis is | (iv) λ = `-1/3` |
Explanation:
(a) Given equation is
(2x + 3y + 4) + λ(6x – y + 12) = 0
⇒ (2 + 6λ)x + (3 – λ)y + 4 + 12λ = 0 ......(i)
If equation (i) is parallel to y-axis
Then 3 – λ = 0
⇒ λ = 3
(b) Given lines are
(2x + 3y + 4) + λ(6x – y + 12) = 0 ......(i)
⇒ (2 + 6λ)x + (3 – λ)y + 4 + 12l = 0
Slope = `-((2 + 6lambda)/(3 - lambda))`
Second equation is 7x + y – 4 = 0 ......(ii)
Slope = – 7
If equation (i) and eq. (ii) are perpendicular to each other
∴ `(-)[-((2 + 6lambda)/(3 - lambda))]` = – 1
⇒ `(14 + 42lambda)/(3 - lambda)` = – 1
⇒ 14 + 42λ = – 3 + λ
⇒ 42λ – λ = – 17
⇒ 41λ = – 17
⇒ λ = `- 17/41`
(c) Given equation is (2x + 3y + 4) + l(6x – y + 12) = 0 ......(i)
If equation (i) passes through the given point (1, 2) then
(2 × 1 + 3 × 2 + 4) + λ(6 × 1 – 2 + 12) = 0
⇒ (2 + 6 + 4) + λ(6 – 2 + 12) = 0
⇒ 12 + 16λ = 0
⇒ λ = `(-12)/16 = (-3)/4`
(d) The given equation is (2x + 3y + 4) + l(6x – y + 12) = 0
⇒ (2 + 6λ)x + (3 – λ)y + 4 + 12λ = 0 ......(i)
If equation (i) is parallel to x-axis, then
2 + 6λ = 0
⇒ λ = `(-1)/3`
APPEARS IN
RELATED QUESTIONS
Find the distance of the point (–1, 1) from the line 12(x + 6) = 5(y – 2).
Find the distance between parallel lines l (x + y) + p = 0 and l (x + y) – r = 0
Find the direction in which a straight line must be drawn through the point (–1, 2) so that its point of intersection with the line x + y = 4 may be at a distance of 3 units from this point.
Find the distance of the point (2, 5) from the line 3x + y + 4 = 0 measured parallel to a line having slope 3/4.
Find the distance of the point (2, 5) from the line 3x + y + 4 = 0 measured parallel to the line 3x − 4y+ 8 = 0.
Find the equation of a line perpendicular to the line \[\sqrt{3}x - y + 5 = 0\] and at a distance of 3 units from the origin.
Show that the perpendiculars let fall from any point on the straight line 2x + 11y − 5 = 0 upon the two straight lines 24x + 7y = 20 and 4x − 3y − 2 = 0 are equal to each other.
Find the perpendicular distance from the origin of the perpendicular from the point (1, 2) upon the straight line \[x - \sqrt{3}y + 4 = 0 .\]
If sum of perpendicular distances of a variable point P (x, y) from the lines x + y − 5 = 0 and 3x − 2y + 7 = 0 is always 10. Show that P must move on a line.
Determine the distance between the pair of parallel lines:
y = mx + c and y = mx + d
Find the ratio in which the line 3x + 4y + 2 = 0 divides the distance between the line 3x + 4y + 5 = 0 and 3x + 4y − 5 = 0
Find the equations of the lines through the point of intersection of the lines x − y + 1 = 0 and 2x − 3y+ 5 = 0, whose distance from the point(3, 2) is 7/5.
Write the value of θ ϵ \[\left( 0, \frac{\pi}{2} \right)\] for which area of the triangle formed by points O (0, 0), A (a cos θ, b sin θ) and B (a cos θ, − b sin θ) is maximum.
The area of a triangle with vertices at (−4, −1), (1, 2) and (4, −3) is
Distance between the lines 5x + 3y − 7 = 0 and 15x + 9y + 14 = 0 is
The value of λ for which the lines 3x + 4y = 5, 5x + 4y = 4 and λx + 4y = 6 meet at a point is
The ratio in which the line 3x + 4y + 2 = 0 divides the distance between the line 3x + 4y + 5 = 0 and 3x + 4y − 5 = 0 is
A plane passes through (1, - 2, 1) and is perpendicular to two planes 2x - 2y + z = 0 and x - y + 2z = 4. The distance of the plane from the point (1, 2, 2) is ______.
The shortest distance between the lines
`bar"r" = (hat"i" + 2hat"j" + hat"k") + lambda (hat"i" - hat"j" + hat"k")` and
`bar"r" = (2hat"i" - hat"j" - hat"k") + mu(2hat"i" + hat"j" + 2hat"k")` is
If P(α, β) be a point on the line 3x + y = 0 such that the point P and the point Q(1, 1) lie on either side of the line 3x = 4y + 8, then _______.
The distance of the point P(1, – 3) from the line 2y – 3x = 4 is ______.
Find the points on the line x + y = 4 which lie at a unit distance from the line 4x + 3y = 10.
If the sum of the distances of a moving point in a plane from the axes is 1, then find the locus of the point.
The distance of the point of intersection of the lines 2x – 3y + 5 = 0 and 3x + 4y = 0 from the line 5x – 2y = 0 is ______.
Find the length of the perpendicular drawn from the point P(3, 2, 1) to the line `overliner = (7hati + 7hatj + 6hatk) + λ(-2hati + 2hatj + 3hatk)`
