Advertisements
Advertisements
प्रश्न
Show that the equation 4x2 + 4xy + y2 – 6x – 3y – 4 = 0 represents a pair of parallel lines. Find the distance between them
Advertisements
उत्तर
The given equation of pair of straight lines is
4x2 + 4xy + y2 – 6x – 3y – 4 = 0 ......(1)
4x2 + 4xy + y2 = (2x + y)2
Let the separate equation of the lines be
2x + y + l = 0 ......(2)
2x + y + m = 0 ......(3)
4x2 + 4xy + y2 – 6x – 3y – 4 = (2x + y + l)(2x + y + m)
Comparing the coefficients of x, y and constant terms on both sides we have
2l + 2m = – 6
l + m = – 3 ......(4)
l + m = – 3 ......(5)
l m = – 4 ......(6)
(l – m)2 = (l + m)2 – 4lm
(l – m )2 = (– 3)2 – 4 × – 4
(l – m)2 = 9 + 16 = 25
l – m = 5 ………… (7)
Solving equations (4) and (7)
(4) ⇒ l + m = – 3
(7) ⇒ l – m = 5
2l + 0 = 2
⇒ l = 1
4) ⇒ l + m = – 3
⇒ m = – 4
∴ The separate equation of the straight lines are
2x + y + 1 =0 and 2x + y – 4 = 0
The distance between the parallel lines is
D = `(- 4 - 1)/sqrt(2^2 + 1^2)`
= `(- 5)/sqrt(4 + 1)`
D = `5/sqrt(5)`
= `sqrt(5)`
∴ The given equation represents a pair of parallel straight lines and the distance between the parallel lines is `sqrt(5)` units.
APPEARS IN
संबंधित प्रश्न
If the equation ax2 + 5xy – 6y2 + 12x + 5y + c = 0 represents a pair of perpendicular straight lines, find a and c.
Show that the equation 12x2 – 10xy + 2y2 + 14x – 5y + 2 = 0 represents a pair of straight lines and also find the separate equations of the straight lines.
Show that the pair of straight lines 4x2 + 12xy + 9y2 – 6x – 9y + 2 = 0 represents two parallel straight lines and also find the separate equations of the straight lines.
Combined equation of co-ordinate axes is:
Prove that the equation to the straight lines through the origin, each of which makes an angle α with the straight line y = x is x2 – 2xy sec 2α + y2 = 0
Find the separate equation of the following pair of straight lines
3x2 + 2xy – y2 = 0
Find the separate equation of the following pair of straight lines
6(x – 1)2 + 5(x – 1)(y – 2) – 4(y – 3)2 = 0
The slope of one of the straight lines ax2 + 2hxy + by2 = 0 is three times the other, show that 3h2 = 4ab
A ∆OPQ is formed by the pair of straight lines x2 – 4xy + y2 = 0 and the line PQ. The equation of PQ is x + y – 2 = 0, Find the equation of the median of the triangle ∆ OPQ drawn from the origin O
Find the value of k, if the following equation represents a pair of straight lines. Further, find whether these lines are parallel or intersecting, 12x2 + 7xy − 12y2 − x + 7y + k = 0
For what values of k does the equation 12x2 + 2kxy + 2y2 +11x – 5y + 2 = 0 represent two straight lines
Choose the correct alternative:
If the equation of the base opposite to the vertex (2, 3) of an equilateral triangle is x + y = 2, then the length of a side is
Choose the correct alternative:
The image of the point (2, 3) in the line y = −x is
The distance between the two points A and A' which lie on y = 2 such that both the line segments AB and A'B (where B is the point (2, 3)) subtend angle `π/4` at the origin, is equal to ______.
Let the equation of the pair of lines, y = px and y = qx, can be written as (y – px) (y – qx) = 0. Then the equation of the pair of the angle bisectors of the lines x2 – 4xy – 5y2 = 0 is ______.
The pair of lines represented by 3ax2 + 5xy + (a2 – 2)y2 = 0 are perpendicular to each other for ______.
