Home
Class 12
MATHS
A variable plane passes through a fixe...

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 locus of the centre of the sphere `O A B Ci s a/x+b/y+c/z=2.`

Text Solution

Verified by Experts

Let `(alpha, beta, gamma)` be any point on the locus. Then according to the given condition, `(alpha, beta, gamma)` is the centre of the sphere through the origin. Therefore, its equation is given by
`" "(x-alpha)^(2)+(y-beta)^(2)+(z-gamma)^(2)=(0-alpha)^(2)+(0-beta)^(2)+(0-gamma)^(2)`
`" "x^(2)+y^(2)+z^(2)-2alphax-2betay-2gammaz=0`
To obtain its point of intersection with the x-axis, we put y=0 and z=0, so that
`" "x^(2)-2alphax=0 or x(x-2alpha)=0 or x=0, 2alpha`
Thus, the plane meets the x-axis at O (0, 0, 0) and `A(2alpha, 0, 0)`. Similarly, it meets the y-axis at `O(0, 0, 0) and B(0, 2beta, 0)`, and the z-axis at `O(0, 0, 0) and C(0, 0, 2gamma)`.
The equation of the plane through `A, B and C` is
`" "(x)/(2alpha)+(y)/(2beta)+(z)/(2gamma) =1" "`(intercept form)
Since it passes through `(a, b, c)`, we get
`" "(a)/(2alpha)+(b)/(2beta)+(c)/(2gamma)=1 or (a)/(alpha)+(b)/(beta)+(c)/(gamma)=2`
Hence, locus of `(alpha, beta, gamma)` is `(a)/(x)+(b)/(y)+(c)/(z)=2`
Promotional Banner

Topper's Solved these Questions

  • THREE-DIMENSIONAL GEOMETRY

    CENGAGE PUBLICATION|Exercise CONCEPT APPLICATION EXERCISE 3.1|12 Videos

  • THREE-DIMENSIONAL GEOMETRY

    CENGAGE PUBLICATION|Exercise CONCEPT APPLICATION EXERCISE 3.2|15 Videos

  • THREE DIMENSIONAL GEOMETRY

    CENGAGE PUBLICATION|Exercise All Questions|291 Videos

  • TRIGONOMETRIC EQUATIONS

    CENGAGE PUBLICATION|Exercise Archives (Numerical value type)|4 Videos

Similar Questions

Explore conceptually related problems

A variable plane passes through a fixed point (alpha,beta,gamma) and meets the axes at A ,B ,a n dCdot show that the locus of the point of intersection of the planes through A ,Ba n dC parallel to the coordinate planes is alphax^(-1)+betay^(-1)+gammaz^(-1)=1.

A variable plane passes through a fixed point (a,b,c) and meets the axes at A ,B ,a n dCdot The locus of the point commom to the planes through A ,Ba n dC parallel to the coordinate planes is

A variable plane which is at a constant distance 3p from origin cuts the coordinate axes at A,B,C respectively. Show that locus of the centroid of the triangleABC is 1/x^2+1/y^2+1/z^2= 1/p^2 .

If a circle passes through the point (a, b) and cuts the circle x^2+y^2=K^2 orthogonally then the equation of the locus of its centre is

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.

A plane passes through a fixed point (a,b,c). The locus of the foot of the perpendicular to it from the origin is a sphere of radius

A sphere of constant radius 2k passes through the origin and meets the axes in A ,B ,a n dCdot The locus of a centroid of the tetrahedron O A B C is a. x^2+y^2+z^2=4k^2 b. x^2+y^2+z^2=k^2 c. 2(x^2+y^2+z)^2=k^2 d. none of these

A variable line passes through a fixed point (x_(1),y_(1)) and meets the axes at A and B. If the rectangle OAPB be completed, the locus of P is, (O being the origin of the system of axes)

If a circle passes through the point (a,b) and cuts the circle x ^(2) +y^(2) =p^(2) orthogonally, then the equation of the locus of its centre is-

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 = 4by