6 Questions
What is the definition of a pair of straight lines?
A set of two lines that lie in the same plane and have no points in common.
What is the general equation of a pair of straight lines?
ax² + 2hxy + by² + 2gx + 2fy + c = 0
What is the condition for an equation to represent a pair of straight lines?
The equation should be of the second degree and the coefficient of x², xy, and y² should not be all zero.
How can the angle between the lines be found?
Using the formula: tanθ = |(m1 - m2) / (1 + m1m2)|
What is the equation of the bisector of the acute angle?
(a - b)x + (h - g)y + (f - c) = 0
What is true about the point of intersection of the two lines?
The point of intersection exists if the lines are not parallel.
Study Notes
Definition
A pair of straight lines is a set of two lines that lie in the same plane and have no points in common.
General Equation
The general equation of a pair of straight lines is:
ax² + 2hxy + by² + 2gx + 2fy + c = 0
where a, h, b, g, f, and c are constants.
Conditions for a Pair of Lines
For an equation to represent a pair of straight lines, the following conditions must be met:
- The equation should be of the second degree (i.e., the highest power of x and y should be 2).
- The coefficient of x², xy, and y² should not be all zero.
- The equation should be factorizable into two linear factors.
Angle between the Lines
The angle between the lines can be found using the formula:
tanθ = |(m1 - m2) / (1 + m1m2)|
where m1 and m2 are the slopes of the two lines.
Bisectors of the Lines
The bisectors of the angles between the lines can be found using the following equations:
- Equation of the bisector of the acute angle: (a - b)x + (h - g)y + (f - c) = 0
- Equation of the bisector of the obtuse angle: (a + b)x + (h + g)y + (f + c) = 0
Point of Intersection
The point of intersection of the two lines can be found by solving the equations simultaneously. If the lines are parallel, there is no point of intersection.
Definition of a Pair of Straight Lines
- A pair of straight lines is a set of two lines that lie in the same plane and have no points in common.
General Equation of a Pair of Straight Lines
- The general equation is: ax² + 2hxy + by² + 2gx + 2fy + c = 0, where a, h, b, g, f, and c are constants.
Conditions for a Pair of Lines
- The equation should be of the second degree (highest power of x and y should be 2).
- The coefficient of x², xy, and y² should not be all zero.
- The equation should be factorizable into two linear factors.
Angle between the Lines
- The angle between the lines can be found using the formula: tanθ = |(m1 - m2) / (1 + m1m2)|, where m1 and m2 are the slopes of the two lines.
Bisectors of the Lines
- The bisector of the acute angle has the equation: (a - b)x + (h - g)y + (f - c) = 0.
- The bisector of the obtuse angle has the equation: (a + b)x + (h + g)y + (f + c) = 0.
Point of Intersection
- The point of intersection of the two lines can be found by solving the equations simultaneously.
- If the lines are parallel, there is no point of intersection.
Learn about the definition and general equation of a pair of straight lines, and the conditions for an equation to represent a pair of lines.
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