Three Dimensional Geometry is crucial in Class 12 and competitive exams because it builds core skills for working with planes, lines, spheres, and their relative positions. Questions often test vector methods (direction ratios, cross/dot products), geometric interpretation (distance, perpendicularity, tangency), and algebraic modeling of spatial objects—exactly the skills used in board exams and JEE/NEET problem-solving.
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Marking
Q1. Find the perpendicular distance from the point to the plane .
Q2. The two lines and are skew. The shortest distance between and is
Q3. A plane passes through the line of intersection of the planes and and is perpendicular to the plane . The equation of that plane is
Q4. Let be the sphere and the plane . Consider the statements:
P1: is a circle of radius .
P2: The centre of that circle is the orthogonal projection of the sphere's centre onto .
Which of the following is correct?
Both P1 and P2 are true but P2 does not correctly explain P1.
P1 is true and P2 is false.
Both P1 and P2 are true and P2 correctly explains P1.
P1 is false and P2 is true.
Q5. Let and . How many planes that contain the line are tangent to the sphere ?
Exactly two such planes exist.
No plane containing is tangent to the sphere.
Exactly one such plane exists.
Infinitely many such planes exist.
Q6. Find the perpendicular distance of the point from the plane .
Q7. The skew lines are and . The shortest distance between and is
Q8. Planes and intersect in a line . The point on nearest to the origin is
Q9. A sphere passes through and is tangent to the plane at the point . The equation of the sphere is
Q10. Let and . The pair of points (one on , one on ) at minimum distance from each other is