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If the pair of lines a x^2+2h x y+b y^2+...

If the pair of lines `a x^2+2h x y+b y^2+2gx+2fy+c=0` intersect on the y-axis, then prove that `2fgh=bg^2+c h^2`

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To prove that if the pair of lines \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) intersect on the y-axis, then \( 2fgh = bg^2 + ch^2 \), we can follow these steps: ### Step 1: Understand the Condition for Intersection on the Y-Axis If the lines intersect on the y-axis, it implies that at the point of intersection, the x-coordinate is 0. Therefore, we can substitute \( x = 0 \) into the given equation. ### Step 2: Substitute \( x = 0 \) into the Equation Substituting \( x = 0 \) in the equation \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) gives: \[ ...
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If the pair of lines ax^2 +2hxy+ by^2+ 2gx+ 2fy +c=0 intersect on the y-axis then

If one of the lines denoted by the line pair a x^2+2h x y+b y^2=0 bisects the angle between the coordinate axes, then prove that (a+b)^2=4h^2

Knowledge Check

  • The circle x^(2) + y^(2) + 2g x + 2fy + c = 0 does not intersect the y-axis if

    A
    A. `g^(2) lt c`
    B
    B. ` f^(2) lt c `
    C
    C. ` 4f^(2) lt c`
    D
    D. ` f^(2) lt 4c`

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    If two curves whose equations are ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 and a'x^2 + 2h'xy + b'y^2 + 2g' x + 2f' y + c = 0 intersect in four concyclic point., then prove that a-b/h = a'-b'/h'

    If the equation ax^(2)+2hxy+by^(2)+2gx+2fy+c=0 represents a pair of parallel lines, then

    Statement 1 : If -2h=a+b , then one line of the pair of lines a x^2+2h x y+b y^2=0 bisects the angle between the coordinate axes in the positive quadrant. Statement 2 : If a x+y(2h+a)=0 is a factor of a x^2+2h x y+b y^2=0, then b+2h+a=0

    Statement 1 : If -2h=a+b , then one line of the pair of lines a x^2+2h x y+b y^2=0 bisects the angle between the coordinate axes in the positive quadrant. Statement 2 : If a x+y(2h+a)=0 is a factor of a x^2+2h x y+b y^2=0, then b+2h+a=0 Both the statements are true but statement 2 is the correct explanation of statement 1. Both the statements are true but statement 2 is not the correct explanation of statement 1. Statement 1 is true and statement 2 is false. Statement 1 is false and statement 2 is true.

    Statement 1 : If -h2=a+b , then one line of the pair of lines a x^2+2h x y+b y^2=0 bisects the angle between the coordinate axes in the positive quadrant. Statement 2 : If a x+y(2h+a)=0 is a factor of a x^2+2h x y+b y^2=0, then b+2h+a=0 Both the statements are true but statement 2 is the correct explanation of statement 1. Both the statements are true but statement 2 is not the correct explanation of statement 1. Statement 1 is true and statement 2 is false. Statement 1 is false and statement 2 is true.

    If the equation ax^(2)+2hxy+by^(2)+2gx+2fy+c=0 respresent the pair of parallel straight lines , then prove that h^(2)=abandabc+2fgh-af^(2)-bg^(2)-ch^(2)=0 .

    If the equation ax^2+2hxy+by^2+2gx+2fy+c=0 represents a pair of parallel lines, prove that h=sqrt(ab) and gsqrt(b)=fsqrt(a)or (h=-sqrt(ab)and gsqrt(b)=-fsqrt(a)) . The distance between them is 2sqrt((((g^2-ac))/(a(a+b)))) .

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