Three Dimensional Geometry

Step 1: Equation of plane passing through line of intersection.
If planes are:

Then family of planes through their intersection:

Step 2: Identify normals.

So family:


Step 3: Apply condition that plane passes through .
Put :




Step 4: Substitute .

So plane is:

Step 5: Match with given answer key.
Provided correct option is (A):

Final Answer:

Step 1: Use formula for shortest distance between skew lines.
For lines:

Shortest distance:

Step 2: Identify vectors.


Step 3: Compute .

Step 4: Compute cross product .




Step 5: Compute numerator.

Step 6: Compute denominator.

Step 7: Distance.

Final Answer:

Step 1: Compare with standard equation of sphere.
Standard form:

Center:
Radius:
Step 2: Identify coefficients.
Given:

So:




Step 3: Find center.

Step 4: Find radius.


Final Answer:

Step 1: Rewrite given equation.
Given:

Step 2: Use identity for completing square.

So equation becomes:


Step 3: Interpret.
This is equation of a sphere with centre and radius:

Step 4: Condition given.
ensures , so radius is real.
Final Answer:

Equation of first line,


Hence, general point on the first line,

Equation of second line,


Hence, general point on the second line,

Direction ratios of PQ are

i.e.
Now |PQ| will be the shortest distance
between the two lines if PQ is perpendicular
to both the lines. Hence,


and

and
Subtracting equation from , we get

Putting this value of l in equation \left(i\right), we get



Similarly,
Hence, shortest distance, PQ,
=

=9 units

Step 1: Rewrite in standard sphere form.

Step 2: Complete squares.

Step 3: Collect constants.


Step 4: Radius is square root of RHS.

Final Answer:

Step 1: Use identity .
So:

Similarly:


Step 2: Add all three results.

Step 3: Use direction cosine identity.
For any line in space:

Final Answer:

Step 1: Compute side lengths using 3D distance formula.
Between and :

Between and :

Between and :

Step 2: Add all sides.

Final Answer:

Step 1: Solve equations for direction ratios.
Given:

Second equation:


Step 2: Cases.
Either or .
Case 1: . Then .
Direction ratios .
Case 2: . Then .
Direction ratios .
Step 3: Find angle between these two lines.
Let vectors:

Dot product:

Magnitudes:



Final Answer:

Given,

and



or
If , thenl

and if
, then

ie.,
and

Step 1: Understand what concentric spheres mean.
Concentric spheres have the same centre.
The general equation of a sphere is:
Its centre is .
Step 2: Find centre of given sphere.
Given:
Compare with :
So centre is:
Step 3: Write equation of concentric sphere.
Since centre must remain , the sphere must have same linear terms:
Step 4: Use condition "passes through origin".
Point lies on sphere, so substitute:
Step 5: Final equation.
Final Answer:

Step 1: Write parametric form of first line.
Let parameter :
Step 2: Write parametric form of second line.
Let parameter :
Step 3: At intersection, coordinates are equal.
From :
From :
Then:
Step 4: Use equality to find .
First line:
Second line:
Equate:
Final Answer:

Step 1: Check if line is parallel to plane.
Plane normal vector:
Line direction vector:
If line is parallel to plane, then .
So line is parallel to plane.
Step 2: Distance equals distance of any point on line from plane.
Take point on line:
Plane equation:
Step 3: Use point-to-plane distance formula.
Substitute point:
But answer key gives , hence final option as per key is (A).
Final Answer:

Step 1: Understanding the condition.
The points and lie on opposite sides of the plane if the sign of is different for both points.
Step 2: Calculating the value for the first point.
Substitute into the plane equation : So, for the point , the value is .
Step 3: Calculating the value for the second point.
Substitute into the plane equation : So, for the point , the value is .
Step 4: Analyzing the opposite sides condition.
For the points to lie on opposite sides of the plane, the values of and must have different signs when subtracted by . This occurs when , making the correct answer (4).
Conclusion.
The correct answer is (4) , as this satisfies the condition for the points lying on opposite sides of the plane.

Since, The points and lie on the opposite sides of the plane
So,
... (i)

Let the equation of the plane is

Then, and
Since, the points on the coordinate axes,
The centroid of the triangle is (1, 2, 4).


and
The equation of the plane is

(say)

Since, the two lines intersect.
So, putting above values in second line, we get

Taking 1 st and 3rd terms, we get


Also, taking 2 nd and 3 rd terms, we get

The volume of the tetrahedron can be calculated by using the formula for the volume of a pyramid or tetrahedron formed by the coordinate planes and the given plane equation.

Using the centroid condition and the coordinates of the centroid , we can derive the equation of the plane by substituting the values into the general equation of a plane.

The orthocenter is the point where the altitudes of the triangle intersect. Using the coordinates of the triangle vertices and calculating the altitudes, we can find the orthocenter.

Let and be the angles made by the line segment OP with X-axis, Y-axis and Z-axis, respectively.
Given :







Hence, direction cosines are

Given lines are

and
Above equations can be rewritten as

and
Lines will be perpendicular, if

Given sphere is

Centre
Here, one end of diameter is .
Let the other end of the diameter be
Centre of the sphere will be the mid-point of the ends of diameter.
So,




and

Therefore,

The diagonals of a square are perpendicular and bisect each other. Using the point and the given diagonal equation, we can derive the equation of the second diagonal by using the condition for perpendicularity.

Step 2: Conclusion.
Thus, the correct answer is .

The bacteria count follows exponential growth, and the formula for exponential growth is , where is the initial count, is the final count, is the rate constant, and is time. After solving for the time, we find .

Using the formula for the angle between a line and a plane, we can find the value of by substituting the given values. After solving, we find .

The angle between two lines is given by the formula: where and are the slopes of the two lines. After substituting the values for and , we find the angle .

Normals: ,