A tangent plane to the ellipsoid x2/a2 + y2/b2 + z2/c2 = 1 meet the coordinate axes at A, B - Maths - Three Dimensional Geometry

Question

A tangent plane to the ellipsoid x2/a2 +
y2/b2 + z2/c2 = 1 meet
the coordinate axes at A, B and C Find the locus in which centroid
of traingle ABC lies

Manbar Singh · Accepted Answer

The equation of any tangent plane to the ellipsoid x2a2 + y2z2 + z2c2 = 1 islx +my + nz = a2l2 + b2m2 + c2n2    
1Now, 1 meets x-axis where y=z=0Put y = z = 0 in 1, we get    lx =a2l2 + b2m2 + c2n2⇒x = 1la2l2 + b2m2 + c2n2 So, 1 meets X-axis at 1la2l2 + b2m2 + c2n2, 0, 0Similarly 1 meets Y-axis and Z-axis at Q0, 1ma2l2 + b2m2 + c2n2, 0 and R0,0,1na2l2 + b2m2 + c2n2If x1,y1,z1 be the centroid of ∆ABC, thenx1 = 131la2l2 + b2m2 + c2n2 + 0 + 0 = 13la2l2 + b2m2 + c2n2y1 = 13ma2l2 + b2m2 + c2n2z1 = 13na2l2 + b2m2 + c2n2Now, x1 =  13la2l2 + b2m2 + c2n2⇒9l2x12 = a2l2 + b2m2 + c2n2⇒1 a2l2 + b2m2 + c2n2 = 19l2x12⇒9a2l2 a2l2 + b2m2 + c2n2 = 9a2l29l2x12⇒9a2l2 a2l2 + b2m2 + c2n2 = a2x12        
2Similarly,9b2m2 a2l2 + b2m2 + c2n2 = b2y12     
3 9c2n2 a2l2 + b2m2 + c2n2 = c2z12    
4Adding 2, 3 and 4, we get9a2l2 a2l2 + b2m2 + c2n2 + 9b2m2 a2l2 + b2m2 + c2n2 + 9c2n2 a2l2 + b2m2 + c2n2 = a2x12 + b2y12  +c2z12   ⇒9 =a2x12 + b2y12  +c2z12So, the required locus of x1,y1,z1 is,a2x2 + b2y2  +c2z2 = 9

A tangent plane to the ellipsoid x2/a2 + y2/b2 + z2/c2 = 1 meet the coordinate axes at A, B and C. Find the locus in which centroid of traingle ABC lies.

A tangent plane to the ellipsoid x²/a² + y²/b² + z²/c² = 1 meet the coordinate axes at A, B and C. Find the locus in which centroid of traingle ABC lies.