Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Homogenizing the given equations
3x2 + 5xy – 3y2 + 2x + 3y = 0 and 3x – 2y – 1 = 0
(i.e) 3x – 2y = 1
We get (3x2 + 5xy – 3y2) + (2x + 3y)(1) = 0
(i.e) (3x2 + 5xy – 3y2) + (2x + 3y)(3x – 2y) = 0
3x2 + 5xy – 3y2 + bx2 – 4xy + 9xy – 6y2 = 0
9x2 + 10xy – 9y2 = 0
Coefficient of x2 + coefficient of y2 = 9 – 9 = 0
⇒ The pair of straight lines are at right angles.
APPEARS IN
संबंधित प्रश्न
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.
Find the angle between the pair of straight lines 3x2 – 5xy – 2y2 + 17x + y + 10 = 0.
Find the combined equation of the straight lines whose separate equations are x − 2y − 3 = 0 and x + y + 5 = 0
Show that 4x2 + 4xy + y2 − 6x − 3y − 4 = 0 represents a pair of parallel lines
Show that 2x2 + 3xy − 2y2 + 3x + y + 1 = 0 represents a pair of perpendicular lines
Find the equation of the pair of straight lines passing through the point (1, 3) and perpendicular to the lines 2x − 3y + 1 = 0 and 5x + y − 3 = 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
Find p and q, if the following equation represents a pair of perpendicular lines
6x2 + 5xy – py2 + 7x + qy – 5 = 0
Show that the equation 9x2 – 24xy + 16y2 – 12x + 16y – 12 = 0 represents a pair of parallel lines. Find the distance between them
Show that the equation 4x2 + 4xy + y2 – 6x – 3y – 4 = 0 represents a pair of parallel lines. Find the distance between them
Prove that one of the straight lines given by ax2 + 2hxy + by2 = 0 will bisect the angle between the coordinate axes if (a + b)2 = 4h2
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
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
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 ______.
