Prove that the product of the perpendiculars from `(alpha,beta)` to the pair of lines `a x^2+2h x y+b y^2=0` is `(aalpha^2-2halphabeta+bbeta^2)/(sqrt((a-b)^2+4h^2))`

Find the equation of pair of lines through the origin and perpendicular to the pair of lines 2x^(2)+11xy+12y^(2)=0

Show that the equation of the pair of lines bisecting the angles between the pair of bisectors of the angles between the pair of lines a x^2+2h x y+b y^2=0 is (a-b)(x^2-y^2)+4h x y=0.

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

Prove that the product of the lengths of the perpendiculars drawn from the points (sqrt(a^(2)-b^(2)), 0) and (-sqrt(a^(2)-b^(2)), 0) to the line (x)/(a)cos theta+(y)/(b)sin theta=1 is b^(2)

Prove that the product of the lengths of the perpendiculars drawn from the points (sqrt(a^(2) - b^(2)) , 0) " and " (- sqrt(a^(2) - b^(2)), 0) to the line x/a cos theta + y/b sin theta = 1 " is " b^(2) .

Prove that the chords of contact of pairs of perpendicular tangents to the ellipse x^2/a^2+y^2/b^2=1 touch another fixed ellipse.

Find the condition for the line y=m x to cut at right angles the conic a x^2+2h x y+b y^2=1.

Find the condition for the line y=m x to cut at right angles the conic a x^2+2h x y+b y^2=1.

If the represented by the equation 3y^2-x^2+2sqrt(3)x-3=0 are rotated about the point (sqrt(3),0) through an angle of 15^0 , on in clockwise direction and the other in anticlockwise direction, so that they become perpendicular, then the equation of the pair of lines in the new position is (a) y^2-x^2+2sqrt(3)x+3=0 (b) y^2-x^2+2sqrt(3)x-3=0 (c) y^2-x^2-2sqrt(3)x+3=0 (d) y^2-x^2+3=0