Advertisements
Advertisements
प्रश्न
Find p and q, if the following equation represents a pair of perpendicular lines
6x2 + 5xy – py2 + 7x + qy – 5 = 0
उत्तर
The equation of the given pair of straight lines is
6x2 + 5xy – py2 + 7x + qy – 5 = 0 ......(1)
Given that equation (1) represents a pair of perpendicular straight lines.
∴ Coefficient of x2 + coefficient of y2 = 0
6 – p = 0
⇒ p = 6
6x2 + 5xy – 6y2 = 6x2 + 9xy – 4xy – 6y2
= 3x(2x + 3y) – 2y (2x + 3y)
= (2x + 3y)(3x – 2y)
Let the separate equation of the straight lines be
2x + 3y + 1 = 0 and 3x – 2y + m = 0
6x2 + 5xy – 6y2 + 7x + qy – 5
= (2x + 3y + 1)(3x – 2y + m)
Comparing the coefficients of x, y and constant terms on both sides
2m + 3l = 7 ......(2)
3m – 2l = q ......(3)
lm = – 5 ......(4)
Equation (4)
⇒ l = 1
m = – 5
or
l = – 1
m = 5
When l = 1
m = – 5 ,
Equation (2) does not satisfy.
∴ l = – 1
m = 5
Substituting in equation (3)
3(5) – 2(–1) = q
⇒ q = 17
∴ The required values are p = 6, q = 17
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:
If the lines 2x – 3y – 5 = 0 and 3x – 4y – 7 = 0 are the diameters of a circle, then its centre is:
Combined equation of co-ordinate axes is:
ax2 + 4xy + 2y2 = 0 represents a pair of parallel lines then ‘a’ is:
Find the combined equation of the straight lines whose separate equations are x − 2y − 3 = 0 and x + y + 5 = 0
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
2x2 – xy – 3y2 – 6x + 19y – 20 = 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
Show that the equation 4x2 + 4xy + y2 – 6x – 3y – 4 = 0 represents a pair of parallel lines. Find the distance between them
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 image of the point (2, 3) in the line y = −x is
Choose the correct alternative:
The area of the triangle formed by the lines x2 – 4y2 = 0 and x = a is
Choose the correct alternative:
If one of the lines given by 6x2 – xy – 4cy2 = 0 is 3x + 4y = 0, then c equals to ______.
Choose the correct alternative:
One of the equation of the lines given by x2 + 2xy cot θ – y2 = 0 is
If `"z"^2/(("z" - 1))` is always real, then z, can lie on ______.
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 ______.
