Advertisements
Advertisements
प्रश्न
For what values of k does the equation 12x2 + 2kxy + 2y2 +11x – 5y + 2 = 0 represent two straight lines
Advertisements
उत्तर
The given equation of the pair of straight line is
12x2 + 2kxy + 2y2 + 11x – 5y + 2 = 0 .......(1)
Compare this equation with the equation
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 .......(2)
a = 12, 2h = 2k, b = 2,
2g = 11, 2f = – 5, c = 2 ,
a =12, h = k, b = 2,
g = `11/2`, f = `- 5/2`, c = 2,
The condition for a second degree equation in x and y to represent a pair of straight lines is
abc + 2fgh – af2 – bg2 – ch2 = 0
`(12)(2)(2) + 2(- 5/2) (11/2)"k" - (12) (- 5/2)^2 - (2) (11/2)^2 - (2)("k")^2` = 0
`48 - (55"k")/2 - 12 xx 25/4 - 2 xx 121/4 - 2"k"^2` = 0
`48 - (55"k")/2 - 75 - 121/2 - 2"k"^2` = 0
96 – 55k – 150 – 121 – 4k2 = 0
– 4k2 – 55k – 175 = 0
4k2 + 55k + 175 = 0
k = `(- 55 +- sqrt(55^2 - 4(4)(175)))/(2(4))`
k = `(- 55 +- sqrt(3025 - 2800))/8`
k = `(- 55 +- sqrt(225))/8`
k = `(- 55 +- 15)/8`
k = `(- 55 + 15)/8` or k = `(- 55 - 15)/8`
k = `(- 40)/8` or k = `(- 70)/8`
k = – 5 or k = `(- 35)/4`
∴ The given equation represents a pair of straight lines when k = – 5 or k = `(- 35)/4`
APPEARS IN
संबंधित प्रश्न
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.
If m1 and m2 are the slopes of the pair of lines given by ax2 + 2hxy + by2 = 0, then the value of m1 + m2 is:
Show that 4x2 + 4xy + y2 − 6x − 3y − 4 = 0 represents a pair of parallel lines
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
6(x – 1)2 + 5(x – 1)(y – 2) – 4(y – 3)2 = 0
Find the separate equation of the following pair of straight lines
2x2 – xy – 3y2 – 6x + 19y – 20 = 0
Show that the equation 4x2 + 4xy + y2 – 6x – 3y – 4 = 0 represents a pair of parallel lines. Find the distance between them
If the pair of straight lines x2 – 2kxy – y2 = 0 bisect the angle between the pair of straight lines x2 – 2lxy – y2 = 0, Show that the later pair also bisects the angle between the former
Prove that the straight lines joining the origin to the points of intersection of 3x2 + 5xy – 3y2 + 2x + 3y = 0 and 3x – 2y – 1 = 0 are at right angles.
Choose the correct alternative:
The coordinates of the four vertices of a quadrilateral are (−2, 4), (−1, 2), (1, 2) and (2, 4) taken in order. The equation of the line passing through the vertex (−1, 2) and dividing the quadrilateral in the equal areas is
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 length of ⊥ from the origin to the line `x/3 - y/4` = 1 is
Choose the correct alternative:
If one of the lines given by 6x2 – xy – 4cy2 = 0 is 3x + 4y = 0, then c equals to ______.
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 ______.
