MathsMediumClass 12

Equation of Line — Symmetric, parametric form

3D Geometry

JEE Qs

Hard

min

Master the parametric form to efficiently represent any point on a line, simplifying problems involving intersections, distances, and conditions on points.

🧮 Key Formulas

Symmetric form of a line passing through (x1, y1, z1) with direction ratios a, b, c: (x - x1)/a = (y - y1)/b = (z - z1)/c

Parametric form of a line: x = x1 + λa, y = y1 + λb, z = z1 + λc (where λ is the parameter)

Vector form of a line: r_vec = a_vec + λb_vec (where a_vec is the position vector of a point on the line and b_vec is the direction vector of the line)

Symmetric form of a line passing through two points (x1, y1, z1) and (x2, y2, z2): (x - x1)/(x2 - x1) = (y - y1)/(y2 - y1) = (z - z1)/(z2 - z1)

✅ Key Points for JEE

1The parametric form (x1+λa, y1+λb, z1+λc) allows representing any arbitrary point on the line, which is critical for solving problems involving distances, intersections, or specific conditions.
2The parameter λ in the parametric form represents a scaled distance along the line from the initial point (x1, y1, z1); if a, b, c are direction cosines, then |λ| is the actual distance.
3To convert between vector form r_vec = a_vec + λb_vec and Cartesian symmetric/parametric forms: a_vec gives the fixed point's coordinates (x1, y1, z1), and b_vec gives the direction ratios (a, b, c).
4If any direction ratio (a, b, or c) is zero, the symmetric form cannot have that term in the denominator. Instead, that numerator term is set to zero (e.g., if a=0, then x - x1 = 0 and (y - y1)/b = (z - z1)/c).
5Direction ratios (a, b, c) define the direction of the line, while direction cosines (l, m, n) are normalized direction ratios where l^2 + m^2 + n^2 = 1. Either can be used, but direction cosines simplify distance calculations.

⚠️ Common Mistakes

✕Incorrectly writing the symmetric form when one or more direction ratios are zero (e.g., dividing by zero instead of equating the numerator to zero).
✕Confusing the coordinates of the fixed point (x1, y1, z1) with the direction ratios (a, b, c) while applying the formulas.
✕Failing to understand that λ is a general parameter, not necessarily the distance from the fixed point unless the direction vector is a unit vector (i.e., a, b, c are direction cosines).
✕Algebraic errors when substituting the parametric point coordinates into equations of other geometric entities (e.g., planes, spheres) to find intersection points.

📝 Practice Questions

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Q4. Let P be the foot of the perpendicular from the point (1, 2, 2) on the line L : x−11 = y+1−1 = z−22 . Let the line →r = (−^i + ^j −2^k) + λ(^i −^j + ^k), λ ∈R, intersect the line L at Q . Then 2(PQ)2 is equal to : (1) 25 (2) 19 (3) 29 (4) 27

2025·MCQMedium

Q16.Let a straight line L pass through the point P(2, −1, 3) and be perpendicular to the lines x−12 = y+11 = z−3−2 and x−3 1 = y−23 = z+24 . If the line L intersects the yz -plane at the point Q , then the distance between the points P and Q is : (1) √10 (2) 2√3 (3) 2 (4) 3

2025·MCQMedium

Q8. Let L1 : x−12 = y−23 = z−34 and L2 : x−23 = y−44 = z−55 be two lines. Then which of the following points lies on the line of the shortest distance between L1 and L2 ? (1) ( 143 , −3, 223 ) (2) (−53 , −7, 1) (3) (2, 3, 13 ) (4) ( 83 , −1, 13 )

2025·MCQHard

Q25.Let L1 : x−13 = y−1−1 = z+10 and L2 : x−22 = 0y = z+4α , α ∈R, be two lines, which intersect at the point B. If P is the foot of perpendicular from the point A(1, 1, −1) on L2 , then the value of 26α( PB)2 is _________

2025·NumericalMedium

Q14.The perpendicular distance, of the line x−1 2 = −1 = z+32 from the point P(2, −10, 1), is : (1) 6 (2) 5√2 (3) 4√3 (4) 3√5

2025·MCQMedium

Q3. Let the position vectors of the vertices A, B and C of a tetrahedron ABCD be ^i + 2^j + ^k,^i + 3^j −2^k and 2^i + ^j −^k respectively. The altitude from the vertex D to the opposite face ABC meets the median line segment through A of the triangle ABC at the point E . If the length of AD is √110 and the volume of the 3 tetrahedron is √805 , then the position vector of E is 6√2 (1) 12 1 (7^i + 4^j + 3^k) (2) 12 (^i + 4^j + 7^k) (3) 1 6 (12^i + 12^j + ^k) (4) 16 (7^i + 12^j + ^k)

2025·Multi conceptHard

NCERT Chapters

Class 12 Maths Ch 11: Three Dimensional Geometry