Distance Formula

Q. Let $f:X\to Y$ be a function such that $f\left(x\right)=\sqrt{x-2}+\sqrt{4-x}$. If $f\left(x\right)$ is both injective as well as surjective, then set $X$ and $Y$ is

1. $[3,4]$ and $[\sqrt{2},2]$
2. $[2,4]$ and $[\sqrt{2},2]$
3. $[2,4]$ and $[1,2]$
4. $[2,3]$ and $[1,2]$

Q.

If $\left(\sin a\right)x^2+\left(\sin a\right)x+(1-\cos a)=0$, then the value of $a$ for which roots of this equation are real and distinct, is

1. $\left(0,2{\tan }^{-1}\frac{1}{4}\right)$

2. $\left(0,\frac{2\pi }{3}\right)$

3. $(0,\pi )$

4. $(0,2\pi )$

Q. If coordinates of two vertices of an equilateral triangle are $(0,0)$ and $(0,2\sqrt{3}),$ then possible coordinates of the third vertex are

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

Q. The polar coordinates of the point whose Cartesian coordinates are $(-1,-\sqrt{3})$, are

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

Q. The locus of the point which is equidistant from the points $A(2,3)$ and $B(-3,4)$ is

1. $5x-y+6=0$
2. $5x+y+6=0$
3. $x+5y+6=0$
4. $x-5y+6=0$

Q. The area (in sq. units) of the triangle formed by the points $(-3,0),(1,-3)$ and $(4,1)$ is

Q.

Find the locus of the point P if $A{P}^2-B{P}^2=18$, where A ≡ (1, 2, –3) and B ≡ (3, –2, 1)

1. 2x + 3y + 4 = 9

2. 2x + 4y + 4z = 9

3. 2x – 3y + 4z = 9

4. 2x – 4y + 4z = 9

Q. The points $(2a,4a),(2a,6a)$ and $(2a+\sqrt{3}a,5a)$ are the vertices of

1. an isosceles triangle
2. an equilateral triangle
3. a right-angled isosceles triangle
4. a scalene triangle

Q. If $A\equiv (3,4),B\equiv (-3,2)$ and $C\equiv (-5,-4)$, then which of the following statements is correct?

1. $ABC$ forms a right angled triangle.
2. $ABC$ forms an acute angled triangle.
3. $ABC$ forms an obtuse angled triangle.
4. $ABC$ does not form a triangle.

Q. If the equation $x^2+y^2=2x$ is written in polar coordinate system, then $r$ can be

1. $0$
2. $1$
3. $2\cos \theta$
4. $2\sin \theta$

Q. The point on $X-$ axis at a distance of $10$ units from $(6,10)$ is

1. $(6,0)$
2. $(10,0)$
3. $(0,0)$
4. $(0,6)$

Q. A circle passes through the points $(2,3)$ and $(4,5)$. If its centre lies on the line, $y-4x+3=0$, then its radius is equal to :

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

Q. The point on the $Y-$axis equidistant from the point $(9,3)$ and $(-5,2),$ is

1. $(0,\frac{31}{2})$
2. $(0,\frac{61}{2})$
3. $(0,-7)$
4. $(0,-9)$

Q. The quadrilateral formed by the points $P(-5,0),Q(-3,-1),R(-2,5)$ and $S(-4,6)$ is a

1. Rectangle
2. Square
3. Parallelogram
4. Rhombus

Q. The length of the latus rectum of a conic is $5$. Its focus is $(-1,1)$ and its directrix is $3x-4y+2=0$, then the conic is

1. a parabola
2. an ellipse
3. a hyperbola
4. a circle

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. 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. If the distance between the points $(5,-2)$ and $(1,a)$ is $5$ units, then the value of $a$ can be

1. $1$
2. $-1$
3. $5$
4. $-5$

Q. The point equidistant from the points (a, 0, 0), (0, b, 0), (0, 0, c) and (0, 0, 0) is

1. $(\frac{a}{3},\frac{b}{3},\frac{c}{3})$
2. $(a,b,c)$
3. $(\frac{a}{2},\frac{b}{2},\frac{c}{2})$
4. $(4a,4b,4c)$

Q.

If the co-ordinates of the points P, Q, R, S be (1, 2, 3), (4, 5, 7), (– 4, 3, – 6) and (2, 0, 2) respectively, then

1. PQ || RS

2. PQ 丄 RS

3. PQ = RS

4. None of these

Q.

Find the locus of the point P if $A{P}^2-B{P}^2=18$, where A ≡ (1, 2, –3) and B ≡ (3, –2, 1)

1. 2x + 3y + 4 = 9

2. 2x + 4y + 4z = 9

3. 2x – 3y + 4z = 9

4. 2x – 4y + 4z = 9

Q. Let the four vertices of a quadrilateral be $A(-3,5),$ $B(3,7),$ $C(6,-2)$ and $D(0,-4)$. Which of the following statements is/are true?

1. Length of one of the diagonals is $\sqrt{130}$ units
2. Area of the quadrilateral is $60$ sq. units
3. The quadrilateral is a rectangle.
4. Both the diagonals of the quadrilateral are equal.

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. If the distance between the points $(5,-2)$ and $(1,a)$ is $5$ units, then the sum of all possible values(s) of $a$ is

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

Q. If $(x,y)$ be the Cartesian coordinates of the point whose polar coordinates are $(\sqrt{2},\frac{\pi }{2})$, then the value of $x^4+y^4$ is

Q. If the Cartesian coordinates of a point are $(-3,-\sqrt{3})$, then the polar coordinates are

1. $(2\sqrt{3},\frac{5\pi }{6})$
2. $(\sqrt{3},\frac{7\pi }{6})$
3. $(2\sqrt{3},\frac{7\pi }{6})$
4. $(2\sqrt{3},\frac{\pi }{6})$

Q.

The coordinates of a point which is equidistant from the points $(0,0,0),(a,0,0),(0,b,0)and(0,0,c)$ are given by _____

1. $(\frac{a}{2},\frac{b}{2},\frac{c}{2})$

2. $(\frac{-a}{2},\frac{-b}{2},\frac{c}{2})$

3. $(\frac{a}{2},\frac{-b}{2},\frac{-c}{2})$

4. $(\frac{-a}{2},\frac{b}{2},\frac{-c}{2})$

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. If $(0,0),(a,2)$ and $(2,b)$ are the vertices of an equilateral triangle such that both $a$ and $b$ lie in $(0,2),$ then the value of $4(a+b)-ab$ is

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

Q.

Distance between the points (1, 3, 2) and (2, 1, 3) is
[MP PET 1988]

1. 12

2. $\sqrt{12}$

3. $\sqrt{6}$

4. 6