If $O(0,0)$,$A(1,2)$,and $B(3,4)$ are the vertices of triangle $OAB$,then the joint equation of the altitude and median drawn from $O$ is

The joint equation of a pair of straight lines passing through the origin and having slopes $(1+\sqrt{2})$ and $\left(\frac{1}{1+\sqrt{2}}\right)$ is $......$

$PQR$ is a right-angled isosceles triangle with the right angle at $P(2, 1)$. If the equation of the line $QR$ is $2x + y = 3$,then the equation representing the pair of lines $PQ$ and $PR$ is:

If the equation of the pair of straight lines passing through the point $(1, 1)$ and perpendicular to the pair of lines $3x^2 + 11xy - 4y^2 = 0$ is $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ (where the constant term is $12$),then find the value of $2(a - h + b - g + f - 12)$.

$A$ pair of straight lines drawn through the origin forms an isosceles triangle right-angled at the origin with the line $2x + 3y = 6$. The area (in sq. units) of the triangle so formed is

$O(0,0), A(1,2), B(3,4)$ are the vertices of $\triangle OAB$. The joint equation of the altitude and median drawn from $O$ is