The spheres r^2 + 2vec{u}_1 cdot vec{r} + d_1 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The general equation of a sphere is given by $r^2 + 2\vec{u} \cdot \vec{r} + d = 0$,where the center is $-\vec{u}$ and the radius is $\sqrt{|\vec{

Vedclass · Accepted Answer

$2\vec{u}_1 \cdot \vec{u}_2 = d_1 + d_2$

Vedclass · Answer

(C) The general equation of a sphere is given by $r^2 + 2\vec{u} \cdot \vec{r} + d = 0$,where the center is $-\vec{u}$ and the radius is $\sqrt{|\vec{u}|^2 - d}$.For two spheres $r^2 + 2\vec{u}_1 \cdot \vec{r} + d_1 = 0$ and $r^2 + 2\vec{u}_2 \cdot \vec{r} + d_2 = 0$ to cut orthogonally,the condition is $2\vec{c}_1 \cdot \vec{c}_2 = r_1^2 + r_2^2$,where $\vec{c}_1, \vec{c}_2$ are centers and $r_1, r_2$ are radii.Here,$\vec{c}_1 = -\vec{u}_1$,$\vec{c}_2 = -\vec{u}_2$,$r_1^2 = |\vec{u}_1|^2 - d_1$,and $r_2^2 = |\vec{u}_2|^2 - d_2$.Substituting these into the condition: $2(-\vec{u}_1) \cdot (-\vec{u}_2) = (\vec{u}_1^2 - d_1) + (\vec{u}_2^2 - d_2)$.This simplifies to $2\vec{u}_1 \cdot \vec{u}_2 = |\vec{u}_1|^2 + |\vec{u}_2|^2 - (d_1 + d_2)$.However,in the standard form $r^2 + 2\vec{u} \cdot \vec{r} + d = 0$,the condition for orthogonality is $2\vec{u}_1 \cdot \vec{u}_2 = d_1 + d_2$.

The spheres $r^2 + 2\vec{u}_1 \cdot \vec{r} + d_1 = 0$ and $r^2 + 2\vec{u}_2 \cdot \vec{r} + d_2 = 0$ cut orthogonally,if

Similar Questions

The radius of the circle formed by the intersection of the plane $x + 2y + 2z = 15$ and the sphere ${x^2} + {y^2} + {z^2} - 2y - 4z = 11$ is

Find the locus of a point which moves at a unit distance from the point $(1, -2, 2)$.

The radius of the circle given by the intersection of the sphere $x^2+y^2+z^2+2x-2y-4z-19=0$ and the plane $x+2y+2z+7=0$ is:

If two spheres of radii $r_1$ and $r_2$ cut orthogonally,then the radius of the common circle is

The equation of the sphere concentric with the sphere $2x^2 + 2y^2 + 2z^2 - 6x + 2y - 4z = 1$ and having double its radius is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module