Perpendicular Distance of a Point from a Line

Q. For a fixed value of $\theta$, if $n_1$ denotes the number of points on the line $3x+4y=5$ which are at a distance of $1+{\sin }^2\theta$ units from $(2,3)$ and $n_2$ denotes the number of points on $3x+4y=5$ which are at a distance of ${\sec }^2\theta +2{\text{cosec}}^2\theta$ units from $(1,3)$, then the value of $n_1+n_2$ is

Q.

Find a point on the $x$-axis, which is equidistant from the points (5, 4) and (2, 3).

1. $(\frac{15}{2},0)$

2. $(\frac{14}{3},0)$

3. $(\frac{27}{7},0)$

4. $(\frac{10}{3},0)$

Q. A variable line passes through a fixed point P, The algebraic sum of the perpendicular distances from (2, 0), (0, 2) and (1, 1) to the line is zero, then the coordinates of the P are

1. (1, -1)
2. (1, 1)
3. (2, 1)
4. (2, 2)

Q.

A point P moves so that its distance from the point (a, 0) is always equal to its distance from the line x + a = 0. The

locus of the point is

1. 

2. 

3. 

4. 

Q.

The coordinates of a point at unit distance from the lines 3x - 4y + 1 = 0 and 3x + 6y + 1 = 0 are

1. $(\frac{6}{5},\frac{-1}{10})$

2. $(0,\frac{-1}{10})$

3. $(\frac{-2}{5},\frac{-13}{10})$

4. $(\frac{-8}{5},\frac{3}{10})$

Q. Let the algebraic sum of the perpendicular distance from the points (2, 0), (0, 2), and (1, 1) to a variable straight line be zero. Then the line passes through a fixed point whose coordinate are ___

1. (1, 1)
2. (2, 2)
3. (0, 0)
4. (3, 3)

Q. The perpendicular bisector of the line segment joining $P(1,4)$ and $Q(k,3)$ has y-intercept $-4$. Then a possible value of $k$ is

1. 2
2. 1
3. -2
4. -4

Q. The point(s) on Y-axis which is/are at a distance of 4 units from the line $\frac{x}{3}+\frac{y}{4}=1$ will be

1. $(0,-\frac{32}{3})$
2. $(0,\frac{8}{3})$
3. $(0,\frac{32}{3})$
4. $(0,-\frac{8}{3})$

Q. The point(s) on x+y=4 that lie(s) at a unit distance from the line 4x+3y-10=0 is/are:

1. (7, -11)
2. (-7, 11)
3. (-3, -1)
4. (3, 1)

Q. The area of an equilateral triangle with the equation of base as x+y-2=0 and the opposite vertex with the coordinates (2, -1) is sq. units.

1. $\frac{1}{\sqrt{3}}$
2. $\frac{1}{2\sqrt{3}}$
3. $\frac{2}{\sqrt{3}}$
4. $\frac{1}{3\sqrt{3}}$