Example 5.
Given $P(2,1,3)$, $Q(6,-5,4)$, $R(2,3,4)$ and $S(-1,5,4)$, determine whether the lines $PQ$ and $RS$ are parallel or skew or intersect.Click to view Text Book Solution
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Example 6.
Given $P(0, 0, 1)$, $Q(3, 6, 4)$, $R(0, 3, 1)$ and $S(3, 0, 4)$, show that the lines $PQ$ and $RS$ are perpendicular.Click to view Text Book Solution
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Example 7.
Find the equation of the line passing through the point $(-4, 7, -3)$ and perpendicular to the line \[ (x, y, z) = (3 + 2k, -1 + 3k, 1 - k). \] Find also the point of intersection of two lines.Click to view Text Book Solution
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Exercise 3.3
Question 1.
Find cos $\angle PAQ$ for the followings.(a) $P(1,2,-1),A(-2,1,5),Q(2,-1,0)$
Short Answer$\frac{4\sqrt{230}}{69}$
(b) $P(0,2,-3),A(2,-1,5),Q(-2,3,-1)$
Short Answer$\frac{34\sqrt{1309}}{1309}$
Question 2.
Determine whether the lines $PQ$ and $RS$ are parallel or skew or intersect. If $PQ$ and $RS$ intersect, are they perpendicular?(a) $P(1,2,3),Q(4,5,6),R(-2,3,5),S(4,9,11)$
Short Answerparallel
(b) $P(3,-1,-3),Q(2,-3,1),R(3,-2,5),S(-1,-2,1)$
Short Answerskew
(c) $P(4,-2,5),Q(-2,6,1),R(-1,1,4),S(3,3,2)$
Short Answerperpendicular
(d) $P(-3,-1,6),Q(-1,3,0),R(0,6,7),S(-4,-4,-1)$
Short AnswerSkew
Question 3.
Find the equation of the line passing through the point (8,-1,-10) and perpendicular to the line $(x,y,z)=(1+2k,2-k,3-7k).$ Find also the point of intersection of two lines.Short Answer\[ \mathbf{(x, y, z) = (8 - 3t,\, -1 + t,\, -10 - t)},(5, 0, -11) \]
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Example 8.
Find the equation of the plane containing $A(1,0,1)$, $B(3,6,4)$ and $C(-2,3,1)$.Click to view Text Book Solution
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Example 9.
Find the equation of the line that passes through the point $(-1, 3, 2)$ and perpendicular to the plane $3x - 2y - z = 3$. Find the point of intersection of the line and the given plane.Click to view Text Book Solution
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Example 10.
Find the equation of the plane containing the point $(-1, 3, 2)$ and parallel to the plane $3x - 2y - 3z = 2$.Click to view Text Book Solution
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Exercise 3.4
Question 1.
Find the equation of the plane containing(a) $A(2, -5, 4) , B(-5, 2, 4) and C(-2, 3, -1)$
Short Answer$5x + 5y + 4z - 1 = 0$
(b) $A(4, 2, -3) , B(1, -2, 4) and C(-1, 0, 3)$
Short Answer10x + 17y + 14z - 32 = 0
Question 2.
Find the equation of the line passing through the point $(3, -2, -2)$ and perpendicular to the plane $-2x + 3y - z = 4$ . Find the point of intersection of the line and the plane.Short Answer\[ x = 3 - 2t, \quad y = -2 + 3t, \quad z = -2 - t \]$(1, 1, -3)$
Question 3.
Find the equation of the plane containing the point $(2, 3, -1)$ and parallel to the plane $-2x + y + 3z = 6.$Short Answer-2x + y + 3z + 4 = 0
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Example 11.
Find the equation of the plane tangent to the sphere \[ (x - 2)^2 + (y - 1)^2 + (z + 1)^2 = 14 \] at the point $(3, 4, 1)$.Click to view Text Book Solution
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Example 12.
Find the equation of the sphere with center $(0, 1, 0)$ and touching the plane $x - 2y + 2z + 5 = 0$.Click to view Text Book Solution
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Example 13.
Find the equation of a sphere that passes through the points $(9, 0, 0)$, $(3, 13, 5)$ and $(11, 0, 10)$, given that its center lies on the $yz$-plane.Click to view Text Book Solution
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**(Useful Formulas)**
Distance Formula:The distance $d$ between two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ is given by: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$
Two-Point Form:
Equation of a line passing through $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$: $$\frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1}$$
- General Equation: $Ax + By + Cz + D = 0$, where $(A, B, C)$ represent the direction ratios of the normal vector.
- One-Point Form: Passing through $(x_1, y_1, z_1)$: $$A(x - x_1) + B(y - y_1) + C(z - z_1) = 0$$
- Intercept Form: $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$
- Normal Form: $lx + my + nz = p$ where $p$ is the distance from the origin.
- Perpendicular Distance: From $(x_0, y_0, z_0)$ to the plane: $$d = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}}$$
Exercise 3.5
Question 1.
Find the equation of sphere with center $C$ and radius $r.$(a) $C( 1, - 2, 4) , r= 3$
Short Answer\[ \mathbf{(x - 1)^2 + (y + 2)^2 + (z - 4)^2 = 9} \]
(b) $C( 2, 6, - 3) , r= 2$
Short Answer\[ \mathbf{(x - 2)^2 + (y -6)^2 + (z +3)^2 = 4} \]
(c) $C(2,3,5),r=5$
Short Answer\[ \mathbf{(x - 2)^2 + (y -3)^2 + (z - 5)^2 = 25} \]
Question 2.
Check whether the given point $P$ lies inside, outside or on a sphere(a) Center $C(0,0,0)$, radius $r=3$ and point $P(1,1,1)$
Short Answerinside
(b) Center $C(0,0,0),$ radius $r=3$ and point $P(2,1,2)$
Short Answeron
(c) Center $C(0,0,0),$ radius $r=3$ and point $P(10,10,10)$
Short Answeroutside
Question 3.
Find the equation of the sphere on the join of (1,-1,1) and (-3,4,5) as diameter.Short Answer \[ \mathbf{x^2 + y^2 + z^2 + 2x - 3y - 6z - 2 = 0} \]
Question 4.
Find the equation of the plane tangent to the sphere $$(x+2)^2+(y-1)^2+(z+3)^2=27$$ at the point $(3,2,-2)$.Short Answer$5x + y + z - 15 = 0$
Question 5.
Find the cquation of the sphere with ccnter $(6,-7,-3)$ and touching the plane $4x-2y-z=17.$Short Answer$(x - 6)^2 + (y + 7)^2 + (z + 3)^2 = \frac{192}{7}$
Question 6.
What is the equation of the sphere which passes through the points $(3,0.2),(-1.1.1) $ and $(2,-5,4)$ and whose center lies on the plane $2x+3y+4z=6?$Short Answer\[ \mathbf{x^2 + (y + 2)^2 + (z - 3)^2 = 14} \]
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