If a point $P$ moves such that its distances from the point $A(1,1)$ and the line $x+y+2=0$ are equal,then the locus of $P$ is

Honey bees also produce beeswax,which finds many uses in industry,such as:

Using the Trapezoidal rule,find the approximate value of $\int_1^4 y \, dx$ based on the following table:

$x$	$1$	$2$	$3$	$4$
$y$	$0.7111$	$0.7222$	$0.7333$	$0.7444$

(in $.1833$)

$A$ police car with a siren of frequency $8 \, kHz$ is moving with uniform velocity $36 \, km/hr$ towards a tall building which reflects the sound waves. The speed of sound in air is $320 \, m/s$. The frequency of the siren heard by the car driver is .... $kHz$.

If the vertex of the conic $y^{2}-4y=4x-4a$ always lies between the straight lines $x+y=3$ and $2x+2y-1=0$,then

If $\int f(x) \sin x \cos x \, dx = \frac{1}{2(b^2 - a^2)} \log f(x) + c$,where $c$ is the constant of integration,then $f(x)$ is equal to