Home
Class 12
MATHS
A plane passes through a fixed point ...

A plane passes through a fixed point `(a ,b ,c)dot` Show that the locus of the foot of the perpendicular to it from the origin is the sphere `x^2+y^2+z^2-a x-b y-c z=0.`

Text Solution

Verified by Experts


Variable plane is passing through the point `A(a, b, c)` and foot of perpendicular from origin on this variable plane is `M(alpha, beta, gamma)`.
We have to find the locus of point `M`.
As shown in the figure, vectors `vec(AM) and vec(OM)` are perpendicular. Thus,
`" "vec(AM)*vec(OM)=0`
`rArr" "(alpha-a)alpha+(beta-b)beta+ (gamma-c)gamma=0`
Therefore, locus of `M` is
`" "x(x-a)+y(y-b)+z(z-c)=0`
Promotional Banner

Topper's Solved these Questions

  • THREE-DIMENSIONAL GEOMETRY

    CENGAGE|Exercise Exercise (Subjective)|16 Videos

  • THREE-DIMENSIONAL GEOMETRY

    CENGAGE|Exercise SUBJECTIVE TYPE|1 Videos

  • THREE-DIMENSIONAL GEOMETRY

    CENGAGE|Exercise Exercise 3.3|19 Videos

  • THREE DIMENSIONAL GEOMETRY

    CENGAGE|Exercise Question Bank|12 Videos

  • TRIGNOMETRIC RATIOS IDENTITIES AND TRIGNOMETRIC EQUATIONS

    CENGAGE|Exercise Question Bank|4 Videos

Similar Questions

Explore conceptually related problems

A straight line passes through a fixed point (6, 8). Find the locus of the foot of the perpendicular on it drawn from the origin is.

A straight line passes through a fixed point (6,8) : Find the locus of the foot of the perpendicular drawn to it from the origin O.

A plane passes through a fixed point (a ,b ,c)dot The locus of the foot of the perpendicular to it from the origin is a sphere of radius a. 1/2sqrt(a^2+b^2+c^2) b. sqrt(a^2+b^2+c^2) c. a^2+b^2+c^2 d. 1/2(a^2+b^2+c^2)

A plane passes through a fixed point (a ,b ,c)dot The locus of the foot of the perpendicular to it from the origin is a sphere of radius a. 1/2sqrt(a^2+b^2+c^2) b. sqrt(a^2+b^2+c^2) c. a^2+b^2+c^2 d. 1/2(a^2+b^2+c^2)

Find the coordinates of the foot of the perpendicular P from the origin to the plane 2x-3y+4z-6=0

The locus of the foot of the perpendicular from the origin on each member of the family (4a+ 3)x - (a+ 1)y -(2a+1)=0

Find the length and the foot of the perpendicular from the point (7,14 ,5) to the plane 2x+4y-z=2.

A variable plane passes through a fixed point (a ,b ,c) and cuts the coordinate axes at points A ,B ,a n dCdot Show that eh locus of the centre of the sphere O A B Ci s a/x+b/y+c/z=2.

A variable circle passes through the point A(a ,b) and touches the x-axis. Show that the locus of the other end of the diameter through A is (x-a)^2=4b y .

Find the locus of the foot of perpendicular from the point (2, 1) on the variable line passing through the point (0, 0).