Two ships leave a port at the same time. One of them moves in the direction of $E 50^{\circ} N$ with a speed of $8 \text{ kmph}$ and the other moves in the direction of $S 20^{\circ} E$ with a speed of $12 \text{ kmph}$. Then the distance between the ships at the end of $2 \text{ hours}$ is (in km).

In a circle of diameter $40 \, cm$,the length of a chord is $20 \, cm$. Find the length of the minor arc of the chord.

The area of a circle having the lines $3x - 4y + 4 = 0$ and $6x - 8y - 7 = 0$ as two of its tangents is:

If the circles $(x+a)^2+(y+b)^2=a^2$ and $(x+c)^2+(y+d)^2=d^2$ cut orthogonally,then $b(b-2d) =$

Consider $L_1: 2x + 3y + p - 3 = 0$; $L_2: 2x + 3y + p + 3 = 0$,where $p$ is a real number,and $C: x^2 + y^2 + 6x - 10y + 30 = 0$.
$STATEMENT-1$: If line $L_1$ is a chord of circle $C$,then line $L_2$ is not always a diameter of circle $C$.
$STATEMENT-2$: If line $L_1$ is a diameter of circle $C$,then line $L_2$ is not a chord of circle $C$.

In an acute-angled $\triangle ABC$,the altitudes from $A, B, C$ when extended intersect the circumcircle again at points $A_1, B_1, C_1$ respectively. If $\angle ABC = 45^{\circ}$,then $\angle A_1 B_1 C_1$ equals (in $^{\circ}$)