Advertisements
Advertisements
प्रश्न
Find k, if the sum of the slopes of the lines represented by x2 + kxy − 3y2 = 0 is twice their product.
उत्तर
x2 + kxy − 3y2 = 0 ...[Given]
Comparing the given equation with ax2 + 2hxy + by2 = 0,
We get, a = 1, 2h = k, b = −3.
Let m1 and m2 be the slopes of the lines represented by x2 + kxy − 3y2 = 0
∴ m1 + m2 = `(-2h)/b = (-k)/(-3) = k/3`
And m1m2 = `a/b = -1/3`
Now, m1 + m2 = 2(m1m2) .......(Given)
∴ `k/3 = 2(- 1/3)`
∴ k = −2
APPEARS IN
संबंधित प्रश्न
Find k, the slopes of the lines represented by 3x2 + kxy - y2 = 0 differ by 4.
Find k, the slope of one of the lines given by kx2 + 4xy – y2 = 0 exceeds the slope of the other by 8.
Find the condition that the line 4x + 5y = 0 coincides with one of the lines given by ax2 + 2hxy + by2 = 0
Find the condition that the line 3x + y = 0 may be perpendicular to one of the lines given by ax2 + 2hxy + by2 = 0
If one of the lines given by ax2 + 2hxy + by2 = 0 is perpendicular to px + qy = 0, show that ap2 + 2hpq + bq2 = 0.
Find the combined equation of the pair of lines through the origin and making an equilateral triangle with the line y = 3.
If the slope of one of the lines given by ax2 + 2hxy + by2 = 0 is four times the other, show that 16h2 = 25ab.
If one of the lines given by ax2 + 2hxy + by2 = 0 bisect an angle between the coordinate axes, then show that (a + b)2 = 4h2 .
Show that the following equation represents a pair of line. Find the acute angle between them:
9x2 - 6xy + y2 + 18x - 6y + 8 = 0
Show that the homogeneous equation of degree 2 in x and y represents a pair of lines passing through the origin if h2 − ab ≥ 0.
Which of the following equation does not represent a pair of lines?
The sum of the slopes of the lines represented by 5x2 + 2xy - 3y2 = 0 is ______
If sum of the slopes of the lines given by x2 - 4pxy + 8y2 = 0 is three times their product then p = ______.
The line x - 2y = 0 is perpenrucular to one of the lines given by ax2 + 2hxy + by2 = 0, when ______.
If the sum of the slopes of the lines represented by x2 + kxy + y2 = 0 is four times their product, the value of k is ______.
The equation 3x2 + 10xy + 8y2 = 0 represents ______
The difference between the slopes of lines represented by 6x2 – 5.xy + y2 = 0 is ______.
Find the value of k. if 2x + y = 0 is one of the lines represented by 3x2 + kxy + 2y2 = 0
(x2 – y2)dx + 2xy dy = 0
Which of the following equation does not represent a pair of lines?
Find separate equations of lines represented by x2 – y2 + x + y = 0.
Show that the joint equation of a pair of straight lines through the origin is a homogeneous equation of second degree in x and y.
The sum of the slopes of the lines given by x2 – 2λxy – 7y2 = 0 is 4 times their product, then the value of λ is ______.
