Advertisements
Advertisements
Question
XOZ-plane divides the join of (2, 3, 1) and (6, 7, 1) in the ratio
Options
3 : 7
2 : 7
–3 : 7
–2 : 7
Solution
−3:7
Let A\[\equiv\](2, 3, 1) and B\[\equiv\]Let the line joining A and B be divided by the xz-plane at point P in the ratio\[\lambda: 1\]
Then, we have,
P\[\equiv \left( \frac{6\lambda + 2}{\lambda + 1}, \frac{7\lambda + 3}{\lambda + 1}, \frac{\lambda + 1}{\lambda + 1} \right)\]
Since P lies on the xz-plane, the y-coordinate of P will be zero.
\[\therefore \frac{7\lambda + 3}{\lambda + 1} = 0\]
\[ \Rightarrow 7\lambda + 3 = 0\]
\[ \Rightarrow \lambda = \frac{- 3}{7}\]
Hence, the xz-plane divides AB in the ratio \[-\]3 : 7
APPEARS IN
RELATED QUESTIONS
Name the octants in which the following points lie:
(1, 2, 3), (4, –2, 3), (4, –2, –5), (4, 2, –5), (–4, 2, –5), (–4, 2, 5),
(–3, –1, 6), (2, –4, –7).
Name the octants in which the following points lie:
(7, 4, –3)
Name the octants in which the following points lie:
(–7, 2 – 5)
Find the image of:
(–5, 4, –3) in the xz-plane.
Find the image of:
(5, 2, –7) in the xy-plane.
A cube of side 5 has one vertex at the point (1, 0, –1), and the three edges from this vertex are, respectively, parallel to the negative x and y axes and positive z-axis. Find the coordinates of the other vertices of the cube.
Find the distances of the point P(–4, 3, 5) from the coordinate axes.
The coordinates of a point are (3, –2, 5). Write down the coordinates of seven points such that the absolute values of their coordinates are the same as those of the coordinates of the given point.
Determine the points in zx-plane are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).
Determine the point on z-axis which is equidistant from the points (1, 5, 7) and (5, 1, –4).
Prove that the point A(1, 3, 0), B(–5, 5, 2), C(–9, –1, 2) and D(–3, –3, 0) taken in order are the vertices of a parallelogram. Also, show that ABCD is not a rectangle.
If A(–2, 2, 3) and B(13, –3, 13) are two points.
Find the locus of a point P which moves in such a way the 3PA = 2PB.
Show that the points (a, b, c), (b, c, a) and (c, a, b) are the vertices of an equilateral triangle.
Verify the following:
(0, 7, –10), (1, 6, –6) and (4, 9, –6) are vertices of an isosceles triangle.
Verify the following:
(0, 7, 10), (–1, 6, 6) and (–4, 9, –6) are vertices of a right-angled triangle.
Verify the following:
(–1, 2, 1), (1, –2, 5), (4, –7, 8) and (2, –3, 4) are vertices of a parallelogram.
Find the locus of the points which are equidistant from the points (1, 2, 3) and (3, 2, –1).
Find the equation of the set of the points P such that its distances from the points A(3, 4, –5) and B(–2, 1, 4) are equal.
Write the distance of the point P(3, 4, 5) from z-axis.
Write the coordinates of the foot of the perpendicular from the point (1, 2, 3) on y-axis.
What is the locus of a point for which y = 0, z = 0?
The ratio in which the line joining the points (a, b, c) and (–a, –c, –b) is divided by the xy-plane is
Find the direction cosines of the line passing through the points P(2, 3, 5) and Q(–1, 2, 4).
The x-coordinate of a point on the line joining the points Q(2, 2, 1) and R(5, 1, –2) is 4. Find its z-coordinate.
Find the coordinates of the point where the line through (3, – 4, – 5) and (2, –3, 1) crosses the plane passing through three points (2, 2, 1), (3, 0, 1) and (4, –1, 0)
Find the image of the point (1, 6, 3) in the line `x/1 = (y - 1)/2 = (z - 2)/3`
If α, β, γ are the angles that a line makes with the positive direction of x, y, z axis, respectively, then the direction cosines of the line are ______.
Find the equations of the two lines through the origin which intersect the line `(x - 3)/2 = (y - 3)/1 = z/1` at angles of `pi/3` each.
Two systems of rectangular axis have the same origin. If a plane cuts them at distances a, b, c and a′, b′, c′, respectively, from the origin, prove that
`1/a^2 + 1/b^2 + 1/c^2 = 1/(a"'"^2) + 1/(b"'"^2) + 1/(c"'"^2)`
Find the equations of the line passing through the point (3,0,1) and parallel to the planes x + 2y = 0 and 3y – z = 0.
Show that the points `(hati - hatj + 3hatk)` and `3(hati + hatj + hatk)` are equidistant from the plane `vecr * (5hati + 2hatj - 7hatk) + 9` = 0 and lies on opposite side of it.
If l1, m1, n1 ; l2, m2, n2 ; l3, m3, n3 are the direction cosines of three mutually perpendicular lines, prove that the line whose direction cosines are proportional to l1 + l2 + l3, m1 + m2 + m3, n1 + n2 + n3 makes equal angles with them.
If the directions cosines of a line are k, k, k, then ______.
The area of the quadrilateral ABCD, where A(0, 4, 1), B(2, 3, –1), C(4, 5, 0) and D(2, 6, 2), is equal to ______.
The locus represented by xy + yz = 0 is ______.
The plane 2x – 3y + 6z – 11 = 0 makes an angle sin–1(α) with x-axis. The value of α is equal to ______.
The direction cosines of the vector `(2hati + 2hatj - hatk)` are ______.
The angle between the line `vecr = (5hati - hatj - 4hatk) + lambda(2hati - hatj + hatk)` and the plane `vec.(3hati - 4hatj - hatk)` + 5 = 0 is `sin^-1(5/(2sqrt(91)))`.
The vector equation of the line `(x - 5)/3 = (y + 4)/7 = (z - 6)/2` is `vecr = (5hati - 4hatj + 6hatk) + lambda(3hati + 7hatj - 2hatk)`.
