3D Geometry

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Questions and Answers

What is the condition for two spheres not to meet and lie farther apart?

|C1C2| > r1 + r2

What is the condition for two spheres to touch internally?

|C1C2| = |r1 - r2|

What is the condition for a plane to cut a sphere in a circle?

p < r

What is the equation of a sphere concentric with the sphere x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0?

x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + λ = 0 Signup and view all the answers

What is the condition for a plane to touch a sphere?

p = r Signup and view all the answers

What is the condition for two spheres to cut orthogonally?

uu' + vv' + ww' = d + d' Signup and view all the answers

If two spheres of radii r1 and r2 cut orthogonally, what is the radius of the common circle?

√(r1^2 - r2^2) Signup and view all the answers

What is the condition for a point to lie on the sphere?

The point satisfies the equation of the sphere Signup and view all the answers

What is the equation of the sphere that passes through the points (0, 0, 0), (0, 2, 0), (1, 0, 0), and (0, 0, 4)?

x^2 + y^2 + z^2 + 2ux + 2vy + 2wz - 1 = 0 Signup and view all the answers

What is the angle of intersection of two spheres?

The angle between the tangent planes to the spheres at their point of intersection Signup and view all the answers

What is the centre of the sphere that passes through the points (0, 0, 0), (0, 2, 0), (1, 0, 0), and (0, 0, 4)?

(1/2, 1, 2) Signup and view all the answers

What is the condition for two spheres to be orthogonal?

The angle of intersection is 90 degrees Signup and view all the answers

What is the condition for a sphere and a plane to be tangent?

The centre of the sphere lies on the plane Signup and view all the answers

What is the number of points at which a line meets a sphere?

Two Signup and view all the answers

What is the condition for l, m, n to be the actual d.c.’s of the line?

l^2 + m^2 + n^2 = 1 Signup and view all the answers

If the direction cosines of a line are $\frac{1}{c}, \frac{1}{c}, \frac{1}{c}$, then what is the value of $c$?

$\pm 3$ Signup and view all the answers

What are the direction cosines of a line that makes an angle of 60° with the y-axis and z-axis?

$(1/2, 1/2, 1/2)$ Signup and view all the answers

If the direction ratios of a line are $2, -3, 6$, then what are the direction cosines of the line?

$(2/7, -3/7, 6/7)$ Signup and view all the answers

If a line has direction cosines $l, m, n$ and is directed so that the angle made by it with the positive direction of the x-axis is acute, then what is the sign of $l$?

$l > 0$ Signup and view all the answers

What are the direction ratios of a line that has direction cosines $2/3, -2/3, 1/3$?

$4, -4, -2$ Signup and view all the answers

What is the definition of direction cosines of a line?

The cosines of the angles made by a line with the positive directions of the coordinate axes. Signup and view all the answers

What is the relation between the direction cosines l, m, and n of a line?

l^2 + m^2 + n^2 = 1 Signup and view all the answers

If a line makes angles α, β, and γ with the positive x, y, and z axes, respectively, what are the direction cosines of the line?

cos(α), cos(β), cos(γ) Signup and view all the answers

What are the direction cosines of the x-axis?

(1, 0, 0) Signup and view all the answers

If a point P has coordinates (x, y, z) and OP = r, what are the coordinates of P in terms of the direction cosines l, m, and n?

(lr, mr, nr) Signup and view all the answers

What is the geometric significance of the direction ratios of a line?

They are the scalar multiples of the direction cosines of the line. Signup and view all the answers

If a, b, c are the direction ratios of a line, then what is the relation between a, b, c, and the direction cosines l, m, n of the line?

a = kl, b = km, c = kn, where k is a constant Signup and view all the answers

If a vector r = ai + bj + ck, then what are the direction ratios of the vector r?

a, b, c Signup and view all the answers

What is the condition for direction cosines l, m, n to be unique?

l^2 + m^2 + n^2 = 1 Signup and view all the answers

If P(x1, y1, z1) and Q(x2, y2, z2) are two points, then what are the direction ratios of the line PQ?

x2 - x1, y2 - y1, z2 - z1 Signup and view all the answers

Study Notes

Intersection of a Straight Line and a Sphere

The equation of a sphere and a straight line are: x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0 and x - α = l(r - l), y - β = m(r - l), z - γ = n(r - l) respectively.
Any point on the line is (α + lr, β + mr, γ + nr).
If this point lies on the sphere, then (α + lr)^2 + (β + mr)^2 + (γ + nr)^2 + 2u(α + lr) + 2v(β + mr) + 2w(γ + nr) + d = 0.
This equation is quadratic in r and gives two values of r, which means the line meets the sphere in two points.

Angle of Intersection of Two Spheres

The angle of intersection of two spheres is the angle between the tangent planes to them at their point of intersection.
This angle is also equal to the angle between the radii of the spheres at their point of intersection.
If the angle of intersection of two spheres is a right angle, the spheres are said to be orthogonal.

Important Tips

Two spheres S1 and S2 with centres C1 and C2 and radii r1 and r2 respectively:
- Do not meet and lie farther apart if |C1C2| > r1 + r2.
- Touch internally if |C1C2| = |r1 - r2|.
- Touch externally if |C1C2| = r1 + r2.
- Cut in a circle if |r1 - r2| < |C1C2| < r1 + r2.
- One lies within the other if |C1C2| < |r1 - r2|.

Equation of a Concentric Sphere

Any sphere concentric with the sphere x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0 is x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + λ = 0, where λ is some real number that makes it a sphere.

Condition for Orthogonality of Two Spheres

If the two spheres x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0 and x^2 + y^2 + z^2 + 2u'x + 2v'y + 2w'z + d' = 0 cut orthogonally, then 2uu' + 2vv' + 2ww' = d + d'.
If the spheres x^2 + y^2 + z^2 = a^2 and x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0 cut orthogonally, then d = a^2.

Direction Ratios and Cosines

Direction ratios of a line are three numbers proportional to the direction cosines of the line.
Direction cosines are the cosines of the angles made by the line with the positive direction of the co-ordinate axes.
The direction cosines of the axes of x, y, and z are respectively (1, 0, 0), (0, 1, 0), and (0, 0, 1).
The relation between the direction cosines is l^2 + m^2 + n^2 = 1.

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3D Geometry

Questions and Answers

What is the condition for two spheres not to meet and lie farther apart?

What is the condition for two spheres to touch internally?

What is the condition for a plane to cut a sphere in a circle?

What is the equation of a sphere concentric with the sphere x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0?

What is the condition for a plane to touch a sphere?

What is the condition for two spheres to cut orthogonally?

If two spheres of radii r1 and r2 cut orthogonally, what is the radius of the common circle?

What is the condition for a point to lie on the sphere?

What is the equation of the sphere that passes through the points (0, 0, 0), (0, 2, 0), (1, 0, 0), and (0, 0, 4)?

What is the angle of intersection of two spheres?

What is the centre of the sphere that passes through the points (0, 0, 0), (0, 2, 0), (1, 0, 0), and (0, 0, 4)?

What is the condition for two spheres to be orthogonal?

What is the condition for a sphere and a plane to be tangent?

What is the number of points at which a line meets a sphere?

What is the condition for l, m, n to be the actual d.c.’s of the line?

If the direction cosines of a line are $\frac{1}{c}, \frac{1}{c}, \frac{1}{c}$, then what is the value of $c$?

What are the direction cosines of a line that makes an angle of 60° with the y-axis and z-axis?

If the direction ratios of a line are $2, -3, 6$, then what are the direction cosines of the line?

If a line has direction cosines $l, m, n$ and is directed so that the angle made by it with the positive direction of the x-axis is acute, then what is the sign of $l$?

What are the direction ratios of a line that has direction cosines $2/3, -2/3, 1/3$?

What is the definition of direction cosines of a line?

What is the relation between the direction cosines l, m, and n of a line?

If a line makes angles α, β, and γ with the positive x, y, and z axes, respectively, what are the direction cosines of the line?

What are the direction cosines of the x-axis?

If a point P has coordinates (x, y, z) and OP = r, what are the coordinates of P in terms of the direction cosines l, m, and n?

What is the geometric significance of the direction ratios of a line?

If a, b, c are the direction ratios of a line, then what is the relation between a, b, c, and the direction cosines l, m, n of the line?

If a vector r = ai + bj + ck, then what are the direction ratios of the vector r?

What is the condition for direction cosines l, m, n to be unique?

If P(x1, y1, z1) and Q(x2, y2, z2) are two points, then what are the direction ratios of the line PQ?

Study Notes

Intersection of a Straight Line and a Sphere

Angle of Intersection of Two Spheres

Important Tips

Equation of a Concentric Sphere

Condition for Orthogonality of Two Spheres

Direction Ratios and Cosines

Studying That Suits You

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