Equation of a Straight Line

Q. If the line $y=mx$ bisects the area enclosed by the lines $x=0,$ $y=0,$ $x=\frac{3}{2}$ and the curve $y=1+4x-x^2,$ then $12m$ is equal to

Q. Consider a plane $P:x-2y+3z=7$ and a line $L:\frac{x-1}{7}=\frac{y}{2}=\frac{z+1}{-1}$. $P_1$ is the plane containing the line $L$ and perpendicular to the plane $P$. If $A(a,b,0)$ is a point on the line of intersection of the planes $P$ and $P_1,$ and two points $B(1,0,–1)$ and $C(8,2,–2)$ lie on the line $L$, then

$[$ Here, $Ar\left(△ABC\right)$ denotes area of $△ABC.]$

1. $a-2b=9$
2. $a-2b=7$
3. $Ar\left(△ABC\right)=\frac{27}{2}\sqrt{\frac{3}{7}}$ sq. units
4. $Ar\left(△ABC\right)=9$ sq. units

Q. If the line $x\cos \alpha +y\sin \alpha =p$ intersects the circle $x^2+y^2=4$ at $A$ and $B$ and chord $AB$ makes an angle of ${30}^{\circ }$ at a point on the circumference of circle then $3p^2=$

Q. $L_1$ is a line intersecting $x$ and $y$ axes at $A(a,0)$ and $B(0,b).$ $L_2$ is a line perpendicular to $L_1$ intersecting $x$ and $y$ axes at $C$ and $D$ respectiveley. If the area of the triangle $OCD$ is 4 times the area of triangle $OAB$, then equation of $AD$ is

1. $x+2y=26$
2. $2x+y+2a=0$
3. $x-2y+2b=0$
4. $2x-y-2a=0$

Q. $L_1$ is a line intersecting $x$ and $y$ axes at $A(a,0)$ and $B(0,b).$ $L_2$ is a line perpendicular to $L_1$ intersecting $x$ and $y$ axes at $C$ and $D$ respectiveley. If the area of the triangle $OCD$ is 4 times the area of triangle $OAB$, then equation of $AD$ is

1. $x+2y=26$
2. $2x+y+2a=0$
3. $x-2y+2b=0$
4. $2x-y-2a=0$

Q. $L_1$ is a line intersecting $x$ and $y$ axes at $A(a,0)$ and $B(0,b).$ $L_2$ is a line perpendicular to $L_1$ intersecting $x$ and $y$ axes at $C$ and $D$ respectiveley. If the area of the triangle $OCD$ is 4 times the area of triangle $OAB$, then equation of $AD$ is

1. $x+2y=26$
2. $2x+y+2a=0$
3. $x-2y+2b=0$
4. $2x-y-2a=0$

Q. A line passes through $(2,2)$ and has $x$-intercept and $y$-intercept as $\alpha \text{units}$ and $\beta \text{units}$ respectively. It makes a triangle of area $A$ with co-ordinate axes. Then the quadratic equation whose roots are $\alpha$ and $\beta$ is :
($\alpha >0,\beta >0$)

1. $x^2-2Ax+2A=0$
2. $x^2-Ax+2A=0$
3. $x^2+2Ax+2A=0$
4. $x^2+Ax+2A=0$

Q. Let the line $\frac{x}{a}+\frac{y}{b}=1$ where $ab>0,$ passes through fixed point $P(\alpha ,\beta ),$ where $\alpha \beta >0.$ If the area formed by the line and the coordinate axes is $S,$ then the least value of $S$ can be expressed as $\lambda ab$ where $\lambda$ is