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`

If the pair of lines ax^(2) + 2hxy + by^(2)+ 2gx + 2fy + c = 0 intersect on the y-axis 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 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 gradient of one of the lines x^2+h x y+2y^2=0 is twice that of the other, then h=

If the equation kx^(2) – 2xy - y^(2)– 2x + 2y = 0 represents a pair of lines, then k =

The equation x^(2) + y^(2) + 2gx + 2fy + c=0 reprsents the circle if

If the pair of straight lines a x^2+2h x y+b y^2=0 is rotated about the origin through 90^0 , then find the equations in the new position.

The image of the pair of lines represented by a x^2+2h x y+b y^2=0 by the line mirror y=0 is a x^2-2h x y-b y^2=0 b x^2-2h x y+a y^2=0 b x^2+2h x y+a y^2=0 a x^2-2h x y+b y^2=0

If the equation a x^2-6x y+y^2 +2gx +2fy +c=0 represents a pair of lines whose slopes are m and m^2, then the value(s) of a is/are

Through a point A on the x-axis, a straight line is drawn parallel to the y-axis so as to meet the pair of straight lines a x^2+2h x y+b y^2=0 at B and Cdot If A B=B C , then (a) h^2=4a b (b) 8h^2=9a b (c) 9h^2=8a b (d) 4h^2=a b