Pair of Straight Lines in Analytic Geometry
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Questions and Answers

What is the definition of a pair of straight lines?

  • A set of two lines that lie in different planes and have one point in common.
  • A set of two lines that lie in different planes and have no points in common.
  • A set of two lines that lie in the same plane and have one point in common.
  • A set of two lines that lie in the same plane and have no points in common. (correct)
  • What is the general equation of a pair of straight lines?

  • ax² + 2hxy + by² + 2gx + 2fy + c = 0 (correct)
  • ax + 2hxy + by + 2gx + 2fy + c = 0
  • ax⁴ + 2hxy + by⁴ + 2gx + 2fy + c = 0
  • ax³ + 2hxy + by³ + 2gx + 2fy + c = 0
  • What is the condition for an equation to represent a pair of straight lines?

  • The equation should be of the fourth degree and the coefficient of x⁴, x³y, and xy³ should not be all zero.
  • The equation should be of the third degree and the coefficient of x³, x²y, and xy² should not be all zero.
  • The equation should be of the second degree and the coefficient of x², xy, and y² should not be all zero. (correct)
  • The equation should be of the first degree and the coefficient of x and y should be zero.
  • How can the angle between the lines be found?

    <p>Using the formula: tanθ = |(m1 - m2) / (1 + m1m2)|</p> Signup and view all the answers

    What is the equation of the bisector of the acute angle?

    <p>(a - b)x + (h - g)y + (f - c) = 0</p> Signup and view all the answers

    What is true about the point of intersection of the two lines?

    <p>The point of intersection exists if the lines are not parallel.</p> Signup and view all the answers

    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.

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    Description

    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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