Advertisements
Advertisements
Question
If the line drawn from the point (–2, – 1, – 3) meets a plane at right angle at the point (1, – 3, 3), find the equation of the plane
Advertisements
Solution
Since, the line drawn from the point B(–2, –1, –3) meets a plane a plane at right angle at the point 4(1, –3, 3).
So, the plane passes through the point 4(1, –3, 3)
Also normal to plane is `(vecr - veca) * vecn` = 0
Where `veca = hati - 3hatj + 3hatk`
⇒ `[(xhati + yhatj + zhatk) - (hati - 3hatj + 3hatk)] * (-3hati + 2hatj - 6hatk)` = 0
⇒ `[(x - 1)hati + (y + 3)hatj + (z - 3)hatk] * (-3hati + 2hatj - 6hatk)` = 0
⇒ `-3x + 3 + 2y + 6 - 6z + 18` = 0
∴ 3x – 2y + 6z – 27 = 0
APPEARS IN
RELATED QUESTIONS
Coordinate planes divide the space into ______ octants.
If the origin is the centroid of the triangle PQR with vertices P (2a, 2, 6), Q (–4, 3b, –10) and R (8, 14, 2c), then find the values of a, b and c.
Name the octants in which the following points lie: (5, 2, 3)
Name the octants in which the following points lie:
(2, –5, –7)
Find the image of:
(–2, 3, 4) in the yz-plane.
Determine the points in zx-plane are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).
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.
Verify the following:
(0, 7, –10), (1, 6, –6) and (4, 9, –6) are vertices of an isosceles triangle.
Verify the following:
(5, –1, 1), (7, –4,7), (1, –6,10) and (–1, – 3,4) are the vertices of a rhombus.
Find the locus of the point, the sum of whose distances from the points A(4, 0, 0) and B(–4, 0, 0) is equal to 10.
Write the distance of the point P(3, 4, 5) from z-axis.
Find the point on y-axis which is at a distance of \[\sqrt{10}\] units from the point (1, 2, 3).
Find the point on x-axis which is equidistant from the points A (3, 2, 2) and B (5, 5, 4).
XOZ-plane divides the join of (2, 3, 1) and (6, 7, 1) in the ratio
The perpendicular distance of the point P (6, 7, 8) from xy - plane is
If the direction ratios of a line are 1, 1, 2, find the direction cosines of the line.
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 co-ordinates of the foot of perpendicular drawn from the point A(1, 8, 4) to the line joining the points B(0, –1, 3) and C(2, –3, –1).
Find the image of the point having position vector `hati + 3hatj + 4hatk` in the plane `hatr * (2hati - hatj + hatk)` + 3 = 0.
The coordinates of the foot of the perpendicular drawn from the point (2, 5, 7) on the x-axis are given by ______.
A line makes equal angles with co-ordinate axis. Direction cosines of this line are ______.
If a line makes angles α, β, γ with the positive directions of the coordinate axes, then the value of sin2α + sin2β + sin2γ is ______.
Prove that the lines x = py + q, z = ry + s and x = p′y + q′, z = r′y + s′ are perpendicular if pp′ + rr′ + 1 = 0.
Find the equation of the plane through the points (2, 1, 0), (3, –2, –2) and (3, 1, 7).
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 foot of perpendicular from the point (2,3,–8) to the line `(4 - x)/2 = y/6 = (1 - z)/3`. Also, find the perpendicular distance from the given point to the line.
Find the length and the foot of perpendicular from the point `(1, 3/2, 2)` to the plane 2x – 2y + 4z + 5 = 0.
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.
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.
The plane 2x – 3y + 6z – 11 = 0 makes an angle sin–1(α) with x-axis. The value of α is equal to ______.
The vector equation of the line through the points (3, 4, –7) and (1, –1, 6) is ______.
The line `vecr = 2hati - 3hatj - hatk + lambda(hati - hatj + 2hatk)` lies in the plane `vecr.(3hati + hatj - hatk) + 2` = 0.
The vector equation of the line `(x - 5)/3 = (y + 4)/7 = (z - 6)/2` is `vecr = (5hati - 4hatj + 6hatk) + lambda(3hati + 7hatj - 2hatk)`.
If the foot of perpendicular drawn from the origin to a plane is (5, – 3, – 2), then the equation of plane is `vecr.(5hati - 3hatj - 2hatk)` = 38.
