The line $3x + 2y = 24$ meets $y$-axis at $A$ and $x$-axis at $B$. The perpendicular bisector of $AB$ meets the line through $(0, - 1)$ parallel to $x$-axis at $C$. The area of the triangle $ABC$ is ............... $\mathrm{sq. \, units}$

Let $PQR$ be a right angled isosceles triangle, right angled 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

The triangle formed by ${x^2} - 9{y^2} = 0$ and $x = 4$ is

The sides of a rhombus $ABCD$ are parallel to the lines, $x - y + 2\, = 0$ and $7x - y + 3\, = 0$. If the diagonals of the rhombus intersect at $P( 1, 2)$ and the vertex $A$ ( different from the origin) is on the $y$ axis, then the ordinate of $A$ is

In a right triangle $ABC$, right angled at $A$, on the leg $AC $ as diameter, a semicircle is described. The chord joining $A$ with the point of intersection $D$ of the hypotenuse and the semicircle, then the length $AC$ equals to

If $x^2-y^2+2 h x y+2 g x+2 f y+c=0$ is the locus of a point, which moves such that it is always equidistant from the lines $x+2 y+7=0$ and $2 x-y$ $+8=0$, then the value of $\mathrm{g}+\mathrm{c}+\mathrm{h}-\mathrm{f}$ equals