Equation of a Sphere : Diametric Form

Q.

(i) Find equation of line joining (1, 2) and (3, 6) using determinants

(ii) Find equation of line joining (3, 1) and (9, 3) using determinants

Q.

If r1 and r2 are distances of points on the ellipse 5x²+5y²+6xy-8=0 which are at maximum and minimum distance from the origin, then r1+r2 is equal to?

Q. If (2, 3, 5) is one end of a diameter of the sphere $x^2+y^2+z^2-6x-12y-2z+20=0$ then co-ordinates of the other end of the diameter are

1. (4, 9, 5)
2. (4, 9, -3)
3. (4, 9, 3)
4. (4, 3, -3)
5. (4, 3, 5)

Q.

Find the equation of a sphere, whose end points of a diameter are (0, 0, 0) and (1, 2, 3)

1. $x^2+y^2+z^2-2x-y-3z=0$

2. $x^2+y^2+z^2-3x-2y-z=0$

3. $x^2+y^2+z^2-x-2y-z=0$

4. $x^2+y^2+z^2-x-2y-3z=0$

Q. Find the equation of the sphere having extremities of one of its diameters as the
points (2, 3, 5) and (-4, 7, 11).

1. $x^2+y^2+z^2-2x-10y-16z+68=0$
2. $x^2+y^2+z^2+2x-10y-16z=68$
3. $x^2+y^2+z^2+2x-10y-16z+68=0$
4. $x^2+y^2+z^2+2x+10y+16z+68=0$

Q. Find the equation of the circle with radius 5 whose centre lies on x-axis and passes through the point (2, 3).

Q. Find the area of the region bounded by the curve $a{y}^2=x^3$, the y-axis and the lines y = a and y = 2a.

Q.

Find the equation of the circle passing through (0, 0) and making intercepts a and b on the coordinate axes.

Q. Find the area of the region in the first quadrant enclosed by x-axis, the line x = $\sqrt{3}y$ and the circle x² + y² = 4.

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. Find the area of the region in the first quadrant enclosed by x-axis, the line x = and the circle x² + y² = 4.

Q. Find the length of latus rectum, transverse axis, conjugate axis, vertices , foci, eccentricity and equations of directrices of the hyperbola 9x2-16y2=144

Q. Calculate the area of the region bounded by the parabolas y² = x and x² = y.

Q.

Co-ordinates of a point equidistant from the points (0, 0, 0), (a, 0, 0), (0, b, 0), (0, 0, c) is

1. 
   

2. 
   

3. 
   

4. (a, b, c)

Q. A sphere with centre $(2,3,0)$ and radius $1$ is drawn in the plane $z$ $=$ $0$, equation of the sphere which passes through this sphere and through the point $(1,1,1)$ is

1. $x^2+y^2+z^2-4x-6y+2z+5=0$
2. $x^2+y^2+z^2-4x-6y+7=0$
3. $x^2+y^2+z^2-4x-6y-5z+12=0$
4. $x^2+y^2+z^2+4x+6y-13=0$

Q. Find the area enclosed by the parabolas y = 5x² and y = 2x² + 9.

Q.

A parabola having equation 4y²+12x-20y+67=0 has its origin shifted to a point (h, k) such that the equation becomes the standard equation y²=4aX.

Find h, k

Find vertex latus rectum, focus, axis , equation of directrix in standard form and transform them with respect to original axes.

Q.

Find the equation of a sphere, whose end points of a diameter are (0, 0, 0) and (1, 2, 3)

1. $x^2+y^2+z^2-x-2y-3z=0$

2. $x^2+y^2+z^2-2x-y-3z=0$

3. $x^2+y^2+z^2-3x-2y-z=0$

4. $x^2+y^2+z^2-x-2y-z=0$

Q. The internal centre of similitude of two circles ${(x-3)}^2+{(y-2)}^2=9,{(x+5)}^2+{(y+6)}^2=9$ is:

1. $(-1,-2)$
2. $(-2,-1)$
3. $(3,2)$
4. $(-5,-6)$

Q. If (2, 3, 5) is one end of a diameter of the sphere $x^2+y^2+z^2-6x-12y-2z+20=0$ then co-ordinates of the other end of the diameter are

1. (4, 3, 5)
2. (4, 9, -3)
3. (4, 9, 3)
4. (4, 3, -3)
5. (4, 9, 5)

Q.

(i) Find equation of line joining (1, 2) and (3, 6) using determinants

(ii) Find equation of line joining (3, 1) and (9, 3) using determinants

Q. Find the equation of the sphere having extremities of one of its diameters as the
points (2, 3, 5) and (-4, 7, 11).

1. $x^2+y^2+z^2-2x-10y-16z+68=0$
2. $x^2+y^2+z^2+2x-10y-16z=68$
3. $x^2+y^2+z^2+2x-10y-16z+68=0$
4. $x^2+y^2+z^2+2x+10y+16z+68=0$

Q. Find the area of the region in the first quadrant enclosed by x-axis, the line y = $\sqrt{3}x$ and the circle x² + y² = 16.

Q.

find the equation of parabola whose axis is parallel to y axis and which passes through the points (0, 4) , (1, 9) , ( -2, 6). also find its latus rectum?

Q. If (2, 3, 5) is one end of a diameter of the sphere $x^2+y^2+z^2-6x-12y-2z+20=0$ then co-ordinates of the other end of the diameter are

1. (4, 3, 5)
2. (4, 9, -3)
3. (4, 9, 3)
4. (4, 3, -3)
5. (4, 9, 5)

Q.

Find the equation of a sphere, whose end points of a diameter are (0, 0, 0) and (1, 2, 3)

1. $x^2+y^2+z^2-x-2y-3z=0$

2. $x^2+y^2+z^2-2x-y-3z=0$

3. $x^2+y^2+z^2-3x-2y-z=0$

4. $x^2+y^2+z^2-x-2y-z=0$

Q. Find the area under the curve y = $\sqrt{6x+4}$ above x-axis from x = 0 to x = 2. Draw a sketch of curve also.