Advertisements
Advertisements
Question
Find the equation of the plane through the points (2, 1, –1) and (–1, 3, 4), and perpendicular to the plane x – 2y + 4z = 10.
Solution
We know that, equation of the plane passing through two points (x1, y1, z1) and (x2, y2, z2) with its normal’s direction ratios is
a(x – x1) + b(y – y1) + c(z – z1) = 0 ......(i)
Now, if the plane is passing through two points (2, 1, –1) and (–1, 3, 4) then
a(x2 – x1) + b(y2 – y1) + c(z2 – z1) = 0
a(–1 – 2) + b(3 – 1) + c(4 + 1) = 0
–3a + 2b + 5c = 0 .......(ii)
As the required plane is perpendicular to the given plane x – 2y + 4z = 10, then
1.a – 2.b + 4.c = 10 ......(iii)
On solving (ii) and (iii) we get,
`a/(8 + 10) = (-b)/(-2 - 5) = c/(6 - 2) = lambda`
So, a = 18λ, b = 17λ and c = 4λ
Thus, the required plane is
18λ(x – 2) + 17λ(y – 1) + 4λ(z + 1) = 0
18x – 36 + 17y – 17 + 4z + 4 = 0
⇒ 18x + 17y + 4z – 49 = 0
APPEARS IN
RELATED QUESTIONS
Coordinate planes divide the space into ______ octants.
Name the octants in which the following points lie:
(4, –3, 5)
Name the octants in which the following points lie:
(–5, –4, 7)
Name the octants in which the following points lie:
(–5, –3, –2)
Find the image of:
(–5, 4, –3) in the xz-plane.
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.
Find the points on z-axis which are at a distance \[\sqrt{21}\]from the point (1, 2, 3).
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.
Are the points A(3, 6, 9), B(10, 20, 30) and C(25, –41, 5), the vertices of a right-angled 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.
Find the locus of the points which are equidistant from the points (1, 2, 3) and (3, 2, –1).
Write the distance of the point P (2, 3,5) from the xy-plane.
Find the point on y-axis which is at a distance of \[\sqrt{10}\] units from the point (1, 2, 3).
The coordinates of the foot of the perpendicular from a point P(6,7, 8) on x - axis are
The length of the perpendicular drawn from the point P(a, b, c) from z-axis is
If the direction ratios of a line are 1, 1, 2, find the direction cosines of the line.
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.
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 ______.
If a variable line in two adjacent positions has direction cosines l, m, n and l + δl, m + δm, n + δn, show that the small angle δθ between the two positions is given by δθ2 = δl2 + δm2 + δn2
Find the length and the foot of perpendicular from the point `(1, 3/2, 2)` to the plane 2x – 2y + 4z + 5 = 0.
The locus represented by xy + yz = 0 is ______.
The direction cosines of the vector `(2hati + 2hatj - hatk)` are ______.
The cartesian equation of the plane `vecr * (hati + hatj - hatk)` is ______.
The intercepts made by the plane 2x – 3y + 5z +4 = 0 on the co-ordinate axis are `-2, 4/3, - 4/5`.
The line `vecr = 2hati - 3hatj - hatk + lambda(hati - hatj + 2hatk)` lies in the plane `vecr.(3hati + hatj - hatk) + 2` = 0.
