Relative Position of a Point with Respect to a Line

Q.

If the points (1, 2) and (3, 4) were to be on the same side of the line 3x-5y+a=0 then

1. 7<a<11

2. a=11

3. 7<a<11

4. a<7 or a>11

Q.

Each of the four inequalities given below defines a region in the $XY$ plane. One of these $4$ regions does not have the following property. For any $2$ points $\left(x_{1,}{y}_1\right)$ and $\left(x_{2,}{y}_2\right)$ in the region, the point $[\frac{(x_1+x_2)}{2},\frac{y_1+y_2}{2}]$ is also in the region. The inequality defining this region is

1. $x^2+2y^2\le 1$

2. $Max\{\left|x\right|,\left|y\right|\}\le 1$

3. $x^2-y^2\le 1$

4. $y^2-x\le 0$

Q. If $(a,a^2)$ falls inside the angle made by the lines $y=\frac{x}{2},x>0,$ and $y=3x,x>0,$ then a belongs to

1. $(-3,\frac{1}{2})$
2. $(0,\frac{1}{2})$
3. $(3,\infty )$
4. $(\frac{1}{2},3)$

Q. The range of values of $\theta$ in the interval $(0,\pi )$ such that the points $(3,5)$ and $(\sin \theta ,\cos \theta )$ lie on the same side of the line $x+y-1=0$, is

1. $(0,\frac{\pi }{2})$
2. $(0,\frac{\pi }{4})$
3. $(\frac{\pi }{4},\frac{\pi }{2})$
4. $(\frac{\pi }{4},\frac{3\pi }{4})$

Q.

The following points A(2a, 4a), B(2a, 6a) and C$(2a+\sqrt{3}a,5a)$, (a > 0) are the vertices of

1. An acute angled triangle

2. A right angled triangle

3. An isosceles triangle

4. None of these

Q.

Orthocentre of the triangle formed by the lines x + y = 1 and xy = 0 is

1. (0, 1)

2. (1, 0)

3. (-1, 1)

4. (0, 0)

Q.

If X = {$4^n$ - 3n - 1 : n ∈ N} and Y = { 9(n-1) : n ∈ N}, then

X ∪ Y is equal to

1. X

2. Y

3. N

4. None of these

Q. The position of the point (8, -9) with respect to the lines 2x+3y-4=0 and 6x+9y+8=0 is

1. Point lies on the same side of the lines
2. Point lies on the different sides of the line
3. Point lies on one of the line
4. None of these

Q.

If |z|=2, then the points representing the complex numbers -1+5z will lie on a

1. Straight line

2. Parabola

3. Circle

4. None of these

Q.

The following points A(2a, 4a), B(2a, 6a) and C$(2a+\sqrt{3}a,5a)$, (a > 0) are the vertices of

1. An acute angled triangle

2. A right angled triangle

3. An isosceles triangle

4. None of these

Q. Find perpendicular distance from the origin to the line joining the points $(\cos \theta ,\sin \theta )$ and $(\cos \varphi ,\sin \varphi )$

Q.

The pair of straight lines joining the origin to the points of intersection of the line y = $2\sqrt{2}x+c$ and the circle $x^2+y^2=2$ are at right angles, if

1. 

2. 

3. 

4. 

Q. $(sin\theta ,cos\theta )$ and (3, 2) lies on the same side of the line x+y=1, then $\theta$ lies between

1. $(0,\frac{\pi }{2})$
2. $(0,\pi )$
3. $(\frac{\pi }{4},\frac{\pi }{2})$
4. $(0,\frac{\pi }{4})$

Q. The range of $\alpha$ for which the points $(\alpha ,\alpha +2)$ and $(\frac{3\alpha }{2},{\alpha }^2)$ lie on opposite sides of the line $2x+3y-6=0$ is

1. $(-\infty ,-2)\cup (0,1)$
2. $(-\infty ,-2)\cup (1,\infty )$
3. $(-\infty ,0)\cup (1,2)$
4. $(-1,0)\cup (1,2)$

Q. The points $(x_1,y_1)$ and $(x_2,y_2)$ lie on the same side of $ax+by+c=0$ if $a{x}_1+b{y}_1+c$ and $a{x}_2+b{y}_2+c$ have the same sign.

1. True
2. False

Q.

The points (3, 4) and (-4, 3) lie on the same side of the line 2x - y + 3 = 0.

1. True

2. False

Q. Abscissae and ordinates of $n$ given points are in $A.P.,$ with first term $a$ and common difference $1$ and $2$ respectively. If algebraic sum of perpendiculars drawn from these given points on a variable line which always passes through the point $(\frac{13}{2},11)$ is $0,$ then the value of $a\cdot n$ is

Q. For a point P in the plane, let $d_1$(p) be the distance of the point $P$ from the lines $x-y=0$ and $x+y=0$ respectively. The area of the region $R$ consisting of all points $P$ lying in the first quadrant of the plane and satisfying
$2\le d_1\left(P\right)+d_2\left(P\right)\le 4,$ is ___

Q. For a point P in the plane, let $d_1$(p) be the distance of the point $P$ from the lines $x-y=0$ and $x+y=0$ respectively. The area of the region $R$ consisting of all points $P$ lying in the first quadrant of the plane and satisfying
$2\le d_1\left(P\right)+d_2\left(P\right)\le 4,$ is ___

Q. For points $P=(x_1,y_1)\text{and}Q=(x_2,y_2)$ of the co-ordinate plane, a new distance $d(P,Q)$ is defined by $d(P,Q)=|x_1-x_2|+|y_1-y_2|$. Let $O=(0,0)$ and $A=(3,2)$. Prove that the set of points in the first quadrant which are equidistant (with respect to the new distance) from $O$ and $A$ consinsts of the union of a line segment of finite length and an infinite ray. Sketch this set in a labelled diagram.

Q. If the point $(\alpha ,{\alpha }^2)$ lies between $x+y-2=0$ and $4x+4y=3,$ then the range of $\alpha$ is

1. $(-2,-\frac{3}{2})\cup (\frac{1}{2},1)$
2. $(-2,-\frac{3}{2})\cup (\frac{1}{2},2)$
3. $(2,-\frac{3}{2})\cup (\frac{1}{2},1)$
4. $(-2,-\frac{1}{2})\cup (\frac{1}{2},1)$