Test: Dimensional Geometry - 9

Follow the two steps given below
1. Any straight line parallel to
4x + 3y + 5 = 0
will be
4x + 3y + c = 0
2. If (i) touches the given circle then they intersect at one point only.

The given circle is

Normal at the point,(x', y') is a line passing through it and perpendicular to the tangent at the point.
∴Its equation is given by 

Remark: Let p be the given point. Then
(i) Exactly two tangents to the circle can be drawn if p is an outside point.
(ii) Only one tangent to the circle can be drawn if the point p is on the circle.
(iii) No tangents (or imaginary' tangents) to the circle can be drawn if P is inside the circle.

Remark : 1. The following three equations coincide:
(i) Equation of the tangent to a circle at a point P(x₁, y₁)
(ii) Equation of the chord of contact of the tangents to the circle from an outside point (x₁, y₁)
(iii) Equation of the polar of a point (x,  y) inside or outside the circle (but not on the circle), with repsect to the circle.
Remark: 2. If the point P(x₁, y₁) is on the circle, then its polar coincides with the tangent at that point.
Remark: 3. If the polar of a point P passes through a point Q, then the polar of Q passes through P.

 Let (r, θ) be a point on the circle.
Then the distance between (r, θ) and (R, α) = a

Condition for a line to lie in a plane.
Let the line be

Then the line will lie in the plane if every point on the line lies in the plane.
Taking

This point lies in plane (ii), if it satisfies the equation of the plane.

This equation will be true for every value of r^i.e. every point of line (1) will lie in the plane), if

Thus conditions in (A) are the required conditions.

Notts that
(a) Satisfies both the conditions in (A)
(b) S a tisfies first in (A) and not the second.
(c) .Also satisfies first in (A) and not the second.
(d) .Also satisfies first, in (A) and not the second It. means that the three straight lines in (b), (c) and (d) are perpendicular lo the normal to the plane. In ot her words, these three straight lines are parallel to the plane but do not lie in the point c.

To find the equation of a plane passing through a given point (x₁, y₁, z₁) and a given line 
The equation of the plane through the line (i) is 

Since the plane (ii) passes through the point (x₁, y₁, z₁), therefore

Eliminating a, b, c from equations (ii), (iii) and (iv), we get the required equation of the plane as

To find the equation of the plane containing the lines

The plant: containing (i) is
A ( x - a) + B ( y - b) + C ( z - c) = 0 ...(iii)
where Aa' + Bb‘ + Cc' 3 =0 ...(iv)
Further, since the plane (iii) also contains line (ii),  therefore , the normal to plane (iii) is perpendicular to line (ii).
∴ Aa + Bb + Cc = 0 ...(v)
Eliminating A, B, C from equations (iii), (iv) and (v), we get the required equation of plane 

This is statement (b). The statements (a) and (c) can be deduced from this by simple propert ies of determinants. Thus (a), (b|, (c) are correct.

The number of arbitrary constants in the equation of a straight line in general.
We know that a given line can be regarded as the intersection of any two planes. In particular, we may take- the two planes perpendicular to two of the coordinate planes, say yz and zx planes. These planes are

Equations (i) and (ii) thus represent the given line. Note that equations (i) and (ii) contain four arbitrary constant a, b, c, d

Given point is : (2, 4, -1)
Given line is :

The coordinates of any points on line (i) are
(r - 5, 4r - 3, -9r + 6)
This will be the required foot of perpendicular if the line joining it to the point (2, 4. -1) is perpendicular to the given line.
This requires

Magnitude and equations of Shortest Distance (S.D)
Here we have to find the magnitude and the equation of shorttest distance between two given lines.

*    It
By definition, the line of shortest distance between (i) and (ii) is perpendicular to both (i) and (ii). Therefore, if [λ,μ,ν] an; the d.c.’s of the line of shortest distance we have


Remark: Two lines given by (i) and (ii) will intersect, i.e. these lines will be coplanar if shortest distance between them is zero. Thus the condition for two lines to be coplanar is
 

The given lines are

Let the line of shortest distance meet the line (i) and (ii) in points P and Q respectively. Then the coordinates of P and Q will be

where r₁, and r₂, are such that the line joining P and Q (line of shortest distance) is perpendicular to both (i) and (ii).
The direction ratios of the line PQ are

The coordinates of points P and Q ,so that PQ becomes the line of shortest distance are (3, 8, 3) and (-3, -7, 6) respectively.
The direction ratios of PQ are
6,15,-3 or 2,5,-1
Required equations of the line PQ of shortest distance are given by

 The three planes are

Denote the determinants as follows:
   
Note that Δ is th e coefficients of x, y and z . Δ₁, Δ₂ and Δ₃, are obtained from Δ by replacing its first column, 2nd column and 3rd column by d₁, d₂ and d₃, respectively.
Also note that if the three planes are to intersect in a line, then no two of them are parallel. Under this case, the line of intersection of any two planes will lie on the third plane. Now the line of intersection of (i) and (ii) is given by

Since line (iv) lies on plane (iii), therefore is perpendicuar to the normal to the plane (iii)


Also since line (iv) lies on plane (iii), therefore the coordinates of any point on the line will satisfy the equation of plane (iii), since



Thus (A) and (R) are the required conditions for the three planes to have a common line of intersection.
In a similar way, it can he shown that Δ = 0 and Δ₁ = 0 or Δ = 0 and Δ₂ = 0 are also the required conditions.
∴ (b), (c), (d) are correct.
Remark: To prove that three planes intersect in a line, verify that Δ = 0 and any one: of Δ₁,Δ₂ and Δ₃ becomes zero.