Test: Dimensional Geometry - 10 - Mathematics MCQ

Since the circle (i) passes through the pole (0, 0), therefore equation (i) gives

The given circle is

Remember the equation of the tangent

Proof: let the two circles

Let C₁, and C₂ be the center of S₁ and S₂ resoectivelv. Then

Comments about Radical Axis
1. Definition : The radical axis of two circles is the locus of a point which moves in such a way that the lengths of the tangents drawn from it to the two circles are equal.
2. The radical axis of two circles is perpendicular !<> the line joining their centres.
3. Equation of the Radical Axis
Let [h, k) be the point on the radical axis. Then the lengths of the tangents from (h, k) to the two circles are equal.

All the three statements (a), (b) and (c) arc correct.
Remark: Remember these properties about radical axis.

A family of circles is called a coaxial system of cricles if any two distinct members of the family have the same radical axis. In other words, all the members of the family have a common radical axis

The equations of the planes are

Let us calculate the determinant A of coefficients.

Since A ≠ 0, therefore the planes will intersect at a point.

The equation of the planes are


∴ Required conditions if that
al+bm+cn=0

The given equation is

Note that the equation of a sphere has three characteristics :
1. it is of second degree in x, y, y
2. the coefficients of x², y² and z² and equal
3. the product icrins xy, yzand zx are absent. 

The given sphere is

Radius =1/2 x diameter

The equation of a plane passing through a fixed point (a, b, c) is given by
A ( x - a) + B ( y - b) + C ( z - c) = 0 ...(i)
Let (α,β,γ) be the foot of perpendicular from the origin on the plane given by (i).
Then the point (α,β,γ) lies on (i)
∴ A(α-α) + B(β-b) + C(γ-c) = 0 ...(ii)
Further the line joining (0,0,0) to (α,β,γ) is normal to the plane, therefore

Let the coordinates of the points A,B,C be (a, 0, 0), (0, b, 0) and (0, 0, c) respectively. Then the equation of the circle passing through O, A, ii and C is given by

Let (α,β,γ) be the coordinates of the centroid of tetrahedron OABC. Then

Substituting in (i) for a, b and c, we get
16(α²+β²+γ²) / 4 = 4 k²
or (α²+β²+​γ²) = k²
∴Required locus is given by
x² + y² + z² = k²

Let the equation of the plane be

∴ The coordinates of the points A, B and C will be (p, 0, 0), (0, q, 0) and (0, 0, r) respectively.
The equation of the sphere passing through the point 0, A, B and C will be 
x² - y² + z² - px - qy - rz = 0
If (α,β,γ) arc the coordinates of the centre of this sphere, then

Since plane (i) passes through (a, b, c) therefore