8 Questions
What is the equation that represents both the straight lines (1) and (2)?
(ax + by + c)(a'x + b'y + c') = 0
What type of equation does a homogeneous second degree equation always represent?
A pair of straight lines passing through the origin
In the equation ax^2 + 2hxy + by^2 = b, what do the values of m1 and m2 represent?
The values of m1 and m2 represent the slopes of the two straight lines represented by the equation.
Under what conditions are the straight lines real according to the equation y = -h ± √(h^2 - ab)?
The straight lines are real if h^2 - ab > 0.
What is the equation for the pair of straight lines represented by the equation ax^2 + 2hxy + by^2 = 0?
The pair of straight lines is represented by the equations y = -h ± √(h^2 - ab)x
What is the result of expanding the equation (ax + by + c)(a'x + b'y + c') = 0?
A second degree equation in x and y
What do the coordinates of any point on equation (1) satisfy?
The coordinates of any point on equation (1) satisfy the equation (ax + by + c)(a'x + b'y + c') = 0.
What do the values of m1 and m2 represent in the equation ax^2 + 2hxy + by^2 = b?
The values of m1 and m2 represent the roots of the quadratic equation m^2 - 2hm + b = 0.
Study Notes
Equations of Straight Lines
- The equation ax^2 + 2hxy + by^2 = 0 represents a pair of straight lines.
- The values of m1 and m2 in the equation ax^2 + 2hxy + by^2 = b represent the slopes of the two lines.
Homogeneous Second Degree Equations
- A homogeneous second degree equation always represents a pair of straight lines.
Conditions for Real Straight Lines
- The straight lines are real if y = -h ± √(h^2 - ab) has a real value.
Expanding Equations
- Expanding the equation (ax + by + c)(a'x + b'y + c') = 0 results in a second-degree equation in x and y.
Point Coordinates
- The coordinates of any point on equation (1) satisfy the equation.
Note
- The equation ax^2 + 2hxy + by^2 = 0 is a general equation of a pair of straight lines passing through the origin.
This quiz covers the equation of a pair of straight lines, focusing on the equations of two straight lines and how to derive the equation that represents both lines. Topics include coordinate geometry and algebraic representation of lines.
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