Equation of a Sphere : General Form

Q. The area of the region in the xy-plane defined by the inequalities $x-2y^2\ge 0,1-x-\left|y\right|\ge 0$ is

1. $\frac{1}{3}$
2. $\frac{1}{4}$
3. $\frac{7}{12}$
4. $\frac{1}{2}$

Q.

If the radius of the sphere $x^2+y^2+z^2-2x-4y-6z$ = 0 is r, then find the value of $r^2$

___

Q. The shortest distance from the plane 12x + 4y + 3z = 327 to the sphere $x^2+y^2+z^2+4x-2y-6z=155$ is

1. $11\frac{3}{4}$
2. $13$
3. $39$
4. $26$

Q. The radius of the circle in which the sphere $x^2+y^2+z^2+2x-2y-4z-19=0$ is cut by the plane x +2y +2z +7 =0 is

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

Q. The radius of circular section in which the sphere $|\overrightarrow{r}|=5$ is cut by the plane $\overrightarrow{r}\cdot (\hat{i}+\hat{j}+\hat{k})=3\sqrt{3},$ is equal to

Q. The shortest distance from the plane 12x +4y+3z =327 to the sphere $x^2+y^2+z^2+4x-2y-6z=155$ is

1. 13
2. 26
3. 39
4. $11\frac{4}{13}$

Q.

If the centre of the sphere $x^2+y^2+z^2-2x-4y-6z=0$ is (a, b, c) , find the value of a+b+c

___

Q. Let any tangent plane to the sphere $(x-a{)}^2+(y-b{)}^2+(z-c{)}^2=r^2$ makes intercepts $a,b,c$ with the coordinate axes at $A,B,C$ respectively. If $P$ is the centre of the sphere, then
$($$ar.$ and $vol.$ denote the area and volume respectively$)$

1. $vol.\left(PABC\right)=\frac{abc}{3}$
2. $ar.\left(△ABC\right)=\frac{abc}{r}$
3. $ar.\left(△PAB\right)=\frac{abc}{r}$
4. $vol.\left(PABC\right)=\frac{abc}{6}$

Q. If $A_0,A_1,A_2,A_3,A_4$ and $A_5$ be a regular hexagon inscribed in a circle of unit radius. Then, the product of the lengths of the line segments $A_0{A}_1,A_0{A}_2$ and $A_0{A}_4$ is:

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

Q.

If the radius of the sphere
$x^2+y^2+z^2-2x-4y-6z$ = 0
is r, then find the value of $r^2$

___

Q.

If the centre of the sphere $x^2+y^2+z^2-2x-4y-6z=0$
is (a, b, c) , find the value of a+b+c

___

Q. A cat is situated at point $A$ (10, 6, -4) and a rat is situated at point $B$ (5, 6, 8). The cat is free to move but the rat is always at rest. The minimum distance travelled by cat to catch the rat is:

1. 5 units
2. 12 units
3. 13 units
4. 17 units

Q.

The plane x + 2y - z = 4 cuts the sphere $x^2+y^2+z^2-x+z-2=0$ in a circle of radius
[AIEEE 2005]

1. 2

2. 1

3. 
   

4. 3

Q. The radius of the circle in which the sphere $x^2+y^2+z^2+2x-2y-4z-19=0$ is cut by the plane $x+2y+2z+7=0$ is

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

Q. The center of the circle given by $\overrightarrow{r}\cdot (\hat{i}+2\hat{j}+2\hat{k})=15$ and $|\overrightarrow{r}-(\hat{j}+2\hat{k})|=4$ is

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

Q. The shortest distance from the plane 12x + 4y + 3z = 327 to the sphere $x^2+y^2+z^2+4x-2y-6z=155$ is

1. $11\frac{3}{4}$
2. $13$
3. $39$
4. $26$

Q. A line passes through the centre of a sphere whose radius is 5 and one of the intercept points is $(1,-2,2)$. If the equation of the line is
$\frac{x}{1}=\frac{y}{-2}=\frac{z}{2}$
, then the equation of the sphere can be

1. $x^2+y^2+z^2-\frac{16}{3}x+\frac{32}{3}y-\frac{32}{3}z=-39$
2. $x^2+y^2+z^2+\frac{4}{3}x-\frac{8}{3}y+\frac{8}{3}z-21=0$
3. $x^2+y^2+z^2+\frac{16}{3}x-\frac{32}{3}y+\frac{32}{3}z=-39$
4. $x^2+y^2+z^2-\frac{16}{3}x+\frac{32}{3}y-\frac{32}{3}z=39$

Q.

If the centre of the sphere $x^2+y^2+z^2-2x-4y-6z=0$
is (a, b, c) , find the value of a+b+c

___

Q. Let any tangent plane to the sphere $(x-a{)}^2+(y-b{)}^2+(z-c{)}^2=r^2$ makes intercepts $a,b,c$ with the coordinate axes at $A,B,C$ respectively. If $P$ is the centre of the sphere, then
$($$ar.$ and $vol.$ denote the area and volume respectively$)$

1. $vol.\left(PABC\right)=\frac{abc}{3}$
2. $ar.\left(△ABC\right)=\frac{abc}{r}$
3. $ar.\left(△PAB\right)=\frac{abc}{r}$
4. $vol.\left(PABC\right)=\frac{abc}{6}$

Q. The shortest distance from the plane 12x + 4y + 3z = 327 to the sphere $x^2+y^2+z^2+4x-2y-6z=155$ is

1. $11\frac{3}{4}$
2. $13$
3. $26$
4. $39$

Q. The greatest positive argument of complex number satisfying $|z-4|=Re\left(z\right)$ is

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

Q.

If the radius of the sphere
$x^2+y^2+z^2-2x-4y-6z$ = 0
is r, then find the value of $r^2$

___