Distance Formula

Q.

Determine if the points $(1,5),(2,3),and(-2,-11)$ are collinear.

Q. The distance of line $3y-2z-1=0=3x-z+4$ from the point $(2,-1,6)$ is

1. $2\sqrt{6}$
2. $\sqrt{26}$
3. $4\sqrt{2}$
4. $2\sqrt{5}$

Q.

Find the point on the $x$-axis which is equidistant from $(2,-5)$ and $(-2,9)$

Q. For what values of $k$, the system of linear equations
$x+y+z=22x+y-z=33x+2y+kz=4$
have a unique solution?

Q. The shortest distance between the point $(\frac{3}{2},0)$ and the curve $y=\sqrt{x},(x>0),$ is :

1. $\frac{\sqrt{5}}{2}$
2. $\frac{3}{2}$
3. $\frac{5}{4}$
4. $\frac{\sqrt{3}}{2}$

Q.

The point $\mathrm{P}$ is equidistant from $\mathrm{A}(1,3),\mathrm{B}(-3,5)$ and $\mathrm{C}(5,-1)$, then $\mathrm{PA}$ is equal to

1. $5$

2. $5\sqrt{5}$

3. $25$

4. $5\sqrt{10}$

Q.

The angle between the lines whose direction cosines are proportional to (1, 2, 1) and (2, -3, 6) is

1. 
   

2. 
   

3. 
   

4. 
   

Q.

Let a point $P$ be such that its distance from the point $(5,0)$ is thrice the distance of $P$ from the point $(-5,0)$. If the locus of the point $P$ is a circle of radius $r$, then $4r^2$ is equal to ?

Q.

If algebraic sum of distances of a variable line from points $(2,0),(0,2)$ and $(-2,-2)$ is zero, then the line passes through the fixed point

1. $(-1,–1)$

2. $(1,1)$

3. $(2,2)$

4. $(0,0)$

Q.

Show that the points (0, 7, 10), (-1, 6, 6) and (-4, 9, 6) are the vertices of an isosceles right-angled triangle.

Q.

The points on the line $x+y=4$ which lie at a unit distance from the line $4x+3y=10,$ are

1. $(3,1),(–7,11)$

2. $(3,1),(7,11)$

3. $(–3,1),(–7,11)$

4. $(1,3),(–7,11)$

Q.

The distance of the point $(-3,4)$ from the origin is?

Q.

Find the square root of the surd $9+2\sqrt{14}$

Q. If the cartesian coordinates of a point are $(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}),$ then the polar coordinates are

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

Q.

$\text{Find the locus of a point such that the sum of its distances from (0, 2) and (0, -2) is 6}.$

Q.

For a > b > c > 0, the distance between (1, 1) and the point of intersection of the lines ax + by + c = 0 and bx + ay + c = 0 is less than $2\sqrt{2}$. Then

1. a + b - c > 0

2. a - b + c < 0

3. a - b + c > 0

4. a + b - c < 0

Q.

The focal chord of $y^2=16x$ is tangent to ${(x–6)}^2+y^2=2$, then the possible values of the slope of this chord, are

1. $\left\{–1,1\right\}$

2. $\left\{–2,2\right\}$

3. $\left\{–2,-\frac{1}{2}\right\}$

4. $\left\{2,-\frac{1}{2}\right\}$

Q. The point on $y-$ axis which is equidistant from the points $(12,3)$ and $(-5,10)$ is

1. $(0,-5)$
2. $(0,-2)$
3. $(0,2)$
4. $(0,5)$

Q.

Find the locus of a point equidistant from the point (2, 4) and the y-axis.

Q.

Find the point on y-axis which is at a distance of $\sqrt{10}$ units from the point (1, 2, 3)

Q. The point(s) on the $y-$axis which is (are) at a distance of $13$ units from the point $(-5,7)$, is (are)

1. $(0,19)$
2. $(0,16)$
3. $(0,-5)$
4. $(0,12)$

Q.

The circumcentre of the triangle with vertices $(0,30),(4,0)$ and $(30,0)$ is

1. $(10,10)$

2. $(10,12)$

3. $(12,12)$

4. $(15,15)$

5. $(17,17)$

Q. The equation of the circle which touches x-axis and whose centre is (1, 2), is

1. 
2. 
3. 
4. 

Q. How to find the number of real roots of any equation?

Q. If the point $P(x_1+t(x_2-x_1),y_1+t(y_2-y_1\left)\right)$ divides the line segment joining the points $A(x_1,y_1)$ and $B(x_2,y_2)$ internally, then the range of $t$ is

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

Q.

The point on the line $3x+4y=5$, which is equidistant from $(1,2)$ and $(3,4)$ is

1. $(7,–4)$

2. $(15,–10)$

3. $\left(\frac{1}{7},\frac{8}{7}\right)$

4. $\left(0,\frac{5}{6}\right)$

Q. The point(s) on the $x-$axis which is (are) at a distance of $5$ units from the point $(6,-3),$ is (are)

1. $(2,0)$
2. $(10,0)$
3. $(3,0)$
4. $(5,0)$

Q.

The equations of one of the straight lines which pass through the point $\left(1,3\right)$ and make an angle ${\tan }^{-1}\sqrt{2}$ with the straight line, $y+1=3\sqrt{2}x$ is:

1. $5\sqrt{2}x+4y-15+4\sqrt{2}=0$

2. $4\sqrt{2}x-5y-5+4\sqrt{2}=0$

3. $4\sqrt{2}x+5y-4\sqrt{2}=0$

4. $4\sqrt{2}x+5y-\left(15+4\sqrt{2}\right)=0$

Q.

Are the points A(3, 6, 9), B(10, 20, 30) and C(25, -41, 5), the vertices of a right-angled triangle ?

Q. Let $A=\left[\begin{array}{ccc}x& y& z\\ y& z& x\\ z& x& y\end{array}\right]$, where $x,y$ and $z$ are real numbers such that $x+y+z>0$ a$xyz=2$.
If $A^2=I_3$, then the value of $x^3+y^3+z^3$ is