Straight Lines — NDA Mathematics PYQs

Directions : For the next two (02) items that follow :
The equation of a straight line which passes through the origin and makes an angle $\alpha$ ($0⩽\alpha ⩽90^{\circ }$) with the line $L:x+\sqrt{3}y+3\sqrt{3}=0$ is $x-\sqrt{3}y=0$.

What is the value of $\alpha$ ?

Directions : For the next two (02) items that follow :
The equation of a straight line which passes through the origin and makes an angle $\alpha$ ($0⩽\alpha ⩽90^{\circ }$) with the line $L:x+\sqrt{3}y+3\sqrt{3}=0$ is $x-\sqrt{3}y=0$.

What is the angle made by the line L with positive direction of y-axis ?

Under what condition will the lines $m^{2}x+ny-1=0$ and $n^{2}x-my+2=0$ be perpendicular?

The vertices of a triangle are $A(1,1)$, $B(0,0)$ and $C(2,0)$. The angular bisectors of the triangle meet at $P$. What are the coordinates of $P$?

Let $A(3,-1)$ and $B(1,1)$ be the end points of line segment $AB$. Let $Q$ be the point situated at a distance $\sqrt{2}$ units from $P$ on the perpendicular bisector line of $AB$. What are the possible coordinates of $Q$?

$ABC$ is an equilateral triangle and $AD$ is the altitude on $BC$. If the coordinates of $A$ are $(1,2)$ and that of $D$ are $(-2,6)$, then what is the equation of $BC$?

The system of equations $2x-3y-5=0$, $15y-10x+50=0$

For different values of $m$, the equation $4y=mx-m+2$ represents

The equation of the locus of a point equidistant from the points $(a,b)$ and $(c,d)$ is $(a-c)x+(b-d)y+k=0$. What is the value of $k$?

What is the sum of the intercepts of the line $\frac{x}{a^2}+\frac{y}{b^2}=\frac{2}{a^2+b^2}$ on the coordinate axes?