Three Dimensional Geometry — JEE Main PYQs

354 previous year questions on Three Dimensional Geometry (Mathematics), newest first. Open any question for its complete step-by-step solution.

Let $(α, β, γ)$ be the co-ordinates of the foot of the perpendicular drawn from the point $(5, 4, 2)$ on the line $r = (- \hat{i} + 3 \hat{j} + \hat{k}) + λ (2 \hat{i} + 3 \hat{j} - \hat{k})$ .
Then the length of the projection of the vector …
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Let the line L pass through the point $(- 3, 5, 2)$ and make equal angles with the positive coordinate axes. If the distance of L from the point $(- 2, r, 1)$ is $\frac{14}{3}$ , then the sum of all possible values of $r$ is :
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Let the line $L_{1}$ be parallel to the vector $- 3 \hat{i} + 2 \hat{j} + 4 \hat{k}$ and pass through the point $(2, 6, 7)$ and the line $L_{2}$ be parallel to the vector $2 \hat{i} + \hat{j} + 3 \hat{k}$ and pass through the point $(4, 3, 5)$ . If the line $L_{3}$ is parallel to the vector …
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Let $P (α, β, γ)$ be the point on the line
$\frac{x - 1}{2} = \frac{y + 1}{- 3} = z$ at a distance $414$ from the point $(1, - 1, 0)$ and nearer to the origin. Then the shortest distance, between the lines …
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If the image of the point $P (1, 2, a)$ in the line $\frac{x - 6}{3} = \frac{y - 7}{2} = \frac{7 - z}{2}$ is $Q (5, b, c)$ , then $a^{2} + b^{2} + c^{2}$ is equal to
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Let L be the line $\frac{x + 1}{2} = \frac{y + 1}{3} = \frac{z + 3}{6}$ and let S be the set of all points $(a, b, c)$ on L, whose distance from the line $\frac{x + 1}{2} = \frac{y + 1}{3} = \frac{z - 9}{0}$ along the line L is 7. Then $\sum_{(a, b, c) \in S} (a + b + c)$ is equal to :
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The vertices B and C of a triangle ABC lie on the line $\frac{x}{1} = \frac{1 - y}{- 2} = \frac{z - 2}{3}$ . The coordinates of A and B are $(1, 6, 3)$ and $(4, 9, α)$ respectively and C is at a distance of 10 units from B. The area (in sq. units) of $△ A B C$ is:
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Let the direction cosines of two lines satisfy the equations: $4 ℓ + m - n = 0$ and $2 mn + 10 n ℓ + 3 ℓ m = 0$ .

Then the cosine of the acute angle between these lines is:
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If the image of the point $P (a, 2, a)$ in the line $\frac{x}{2} = \frac{y + a}{1} = \frac{z}{1}$ is Q and the image of Q in the line $\frac{x - 2 b}{2} = \frac{y - a}{1} = \frac{z + 2 b}{- 5}$ is P, then $a + b$ is equal to
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Let the lines $L_{1} : r = \hat{i} + 2 \hat{j} + 3 \hat{k} + λ (2 \hat{i} + 3 \hat{j} + 4 \hat{k})$ , $λ \in R$ and $L_{2} : r = (4 \hat{i} + \hat{j}) + μ (5 \hat{i} + 2 \hat{j} + \hat{k}), μ \in R$ , intersect at the point R . Let P and Q be the points ly …
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Let a line L passing through the point $P (1, 1, 1)$ be perpendicular to the lines $\frac{x - 4}{4} = \frac{y - 1}{1} = \frac{z - 1}{1}$ and $\frac{x - 17}{1} = \frac{y - 71}{1} = \frac{z}{0}$ . Let the line L intersect the yz-plane at the point Q. Another lin …
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The sum of all values of $α$ , for which the shortest distance between the lines
$\frac{x + 1}{α} = \frac{y - 2}{- 1} = \frac{z - 4}{- α}$ and $\frac{x}{α} = \frac{y - 1}{2} = \frac{z - 1}{2 α}$ is $2$ , is
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If the distances of the point $(1, 2, a)$ from the line $\frac{x - 1}{1} = \frac{y}{2} = \frac{z - 1}{1}$ along the lines $L_{1} : \frac{x - 1}{3} = \frac{y - 2}{4} = \frac{z - a}{b}$ and …
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Let $Q (a, b, c)$ be the image of the point $P (3, 2, 1)$ in the line $\frac{x - 1}{1} = \frac{y}{2} = \frac{z - 1}{1}$ . Then the distance of $Q$ from the line $\frac{x - 9}{3} = \frac{y - 9}{2} = \frac{z - 5}{- 2}$ is
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If the distance of the point $P (43, α, β), β < 0$ , from the line $r = 4 \hat{i} - \hat{k} + μ (2 \hat{i} + 3 \hat{k}), μ \in R$ along a line with direction ratios $3, - 1, 0$ is …
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If the point of intersection of the lines $\frac{x + 1}{3} = \frac{y + a}{5} = \frac{z + b + 1}{7}$ and $\frac{x - 2}{1} = \frac{y - b}{4} = \frac{z - 2 a}{7}$ lies on $x y$ -plane, then the value of $a + b$ is :
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Let a line L passing through the point $(1, 1, 1)$ be perpendicular to both the vectors $2 \hat{i} + 2 \hat{j} + \hat{k}$ and $\hat{i} + 2 \hat{j} + \hat{k}$ . If $P (a, b, c)$ is the foot of perpendicular from the origin on the line L, then the value of $34 (a + b + c)$ is :
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The shortest distance between the lines
$r = (\frac{1}{3} \hat{i} + 2 \hat{j} + \frac{8}{3} \hat{k}) + λ (2 \hat{i} - 5 \hat{j} + 6 \hat{k})$

and …
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If $(2 α + 1, α^{2} - 3 α, \frac{α - 1}{2})$ is the image of $(α, 2 α, 1)$ in the line $\frac{x - 2}{3} = \frac{y - 1}{2} = \frac{z}{1}$ , then the possible value(s) of $α$ is(are) :
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A line with direction ratios $1, - 1, 2$ intersects the lines $\frac{x}{2} = \frac{y}{3} = \frac{z + 1}{3}$ and $\frac{x + 1}{- 1} = \frac{y - 2}{1} = \frac{z}{4}$ at the points P and Q, respectively. If the length of the line segment PQ is $α$ , then $225 α^{2}$ is equal to:
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The square of the distance of the point (-2,-8,6) from the line $\frac{x - 1}{1} = \frac{y - 1}{2} = \frac{z}{- 1}$ along the line $\frac{x + 5}{1} = \frac{y + 5}{- 1} = \frac{z}{2}$ is equal to:
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The square of the distance of the point $P (5, 6, 7)$ from the line $\frac{x - 2}{2} = \frac{y - 5}{3} = \frac{z - 2}{4}$ is equal to :
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The square of the distance of the point of intersection of the line $r = (\hat{i} + \hat{j} - \hat{k}) + λ (\overset{a}{^} - \hat{j})$ , $a \neq = 0$ and $r = (4 \hat{i} - \hat{k}) + μ (2 \hat{i} + a \hat{k})$ from t …
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Let a triangle PQR be such that P and Q lie on the line $\frac{x + 3}{8} = \frac{y - 4}{2} = \frac{z + 1}{2}$ and are at a distance of 6 units from $R (1, 2, 3)$ . If $(α, β, γ)$ is the centroid of $Δ PQR$ , then $α + β + γ$ is equal to:
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If the distance of the point $(a, 2, 5)$ from the image of the point $(1, 2, 7)$ in the line $\frac{x}{1} = \frac{y - 1}{1} = \frac{z - 2}{2}$ is 4, then the sum of all possible values of a is equal to:
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The shortest distance between the lines $\frac{x - 4}{1} = \frac{y - 3}{2} = \frac{z - 2}{- 3}$ and $\frac{x + 2}{2} = \frac{y - 6}{4} = \frac{z - 5}{- 5}$ is:
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Let the image of the point $P (0, - 5, 0)$ in the line $\frac{x - 1}{2} = \frac{y}{1} = \frac{z + 1}{- 2}$ be the point R and the image of the point $Q (0, \frac{- 1}{2}, 0)$ in the line $\frac{x - 1}{- 1} = \frac{y + 9}{4} = \frac{z + 1}{1}$ be the …
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Let the image of the point $P (1, 6, a)$ in the line $L : \frac{x}{1} = \frac{y - 1}{2} = \frac{z - a + 1}{b}, b > 0$ , be $(\frac{a}{3}, 0, a + c)$ . If …
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Let a line L be perpendicular to both the lines
$L_{1} : \frac{x + 1}{3} = \frac{y + 3}{5} = \frac{z + 5}{7}$ and
$L_{2} : \frac{x - 2}{1} = \frac{y - 4}{4} = \frac{z - 6}{7}$ .
If $θ$ is the acute angle between the lines L an …
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Let the foot of perpendicular from the point $(λ, 2, 3)$ on the line $\frac{x - 4}{1} = \frac{y - 9}{2} = \frac{z - 5}{1}$ be the point $(1, μ, 2)$ . Then the distance between the lines $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z + 4}{6}$ and …
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Let a line $L_{1}$ pass through the origin and be perpendicular to the lines
$L_{2} : r = (3 + t) \hat{i} + (2 t - 1) \hat{j} + (2 t + 4) \hat{k}$ and …
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Let the point A be the foot of perpendicular drawn from the point $P (a, b, 0)$ on the line $\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - α}{3}$ . If the midpoint of the line segment PA is $(0, \frac{3}{4}, \frac{- 1}{4})$ , then the value of …
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Let $L_{1} : \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ and $L_{2} : \frac{x - 2}{3} = \frac{y - 4}{4} = \frac{z - 5}{5}$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_{1}$ and $L_{2}$ ?
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Let $L_{1} : \frac{x - 1}{3} = \frac{y - 1}{- 1} = \frac{z + 1}{0}$ and $L_{2} : \frac{x - 2}{2} = \frac{y}{0} = \frac{z + 4}{α}, α \in R$ , be two lines, which intersect at the point B. If P is the foot of perp …
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Let a line pass through two distinct points $P (- 2, - 1, 3)$ and Q , and be parallel to the vector $3 \hat{i} + 2 \hat{j} + 2 \hat{k}$ . If the distance of the point $Q$ from the point $R (1, 3, 3)$ is 5 , then the square of the area of $△ PQR$ is equal to:
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The perpendicular distance, of the line $\frac{x - 1}{2} = \frac{y + 2}{- 1} = \frac{z + 3}{2}$ from the point $P (2, - 10, 1)$ , is:
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Let P be the foot of the perpendicular from the point $Q (10, - 3, - 1)$ on the line $\frac{x - 3}{7} = \frac{y - 2}{- 1} = \frac{z + 1}{- 2}$ . Then the area of the right angled triangle PQR , where R is the point $(3, - 2, 1)$ , is
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The distance of the line $\frac{x - 2}{2} = \frac{y - 6}{3} = \frac{z - 3}{4}$ from the point $(1, 4, 0)$ along the line $\frac{x}{1} = \frac{y - 2}{2} = \frac{z + 3}{3}$ is :
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If the square of the shortest distance between the lines $\frac{x - 2}{1} = \frac{y - 1}{2} = \frac{z + 3}{- 3}$ and $\frac{x + 1}{2} = \frac{y + 3}{4} = \frac{z + 5}{- 5}$ is $\frac{m}{n}$ , where $m, n$ are coprime numbers, then $m + n$ is equal to:
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Let a straight line L pass through the point $P (2, - 1, 3)$ and be perpendicular to the lines $\frac{x - 1}{2} = \frac{y + 1}{1} = \frac{z - 3}{- 2}$ and $\frac{x - 3}{1} = \frac{y - 2}{3} = \frac{z + 2}{4}$ . If the line L intersects the yz …
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Let P be the foot of the perpendicular from the point $(1, 2, 2)$ on the line $L : \frac{x - 1}{1} = \frac{y + 1}{- 1} = \frac{z - 2}{2}$ . Let the line …
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Let $a = \hat{i} + 2 \hat{j} + \hat{k}$ and $b = 2 \hat{i} + 7 \hat{j} + 3 \hat{k}$ . Let $L_{1} : r = (- \hat{i} + 2 \hat{j} + \hat{k}) + λ a, λ \in R$ and $L_{2} : r = (\hat{j} + \hat{k}) + μ b, μ \in R$ be two lines. If the line $L_{3}$ passes through the …
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Let $L_{1} : \frac{x - 1}{1} = \frac{y - 2}{- 1} = \frac{z - 1}{2}$ and $L_{2} : \frac{x + 1}{- 1} = \frac{y - 2}{2} = \frac{z}{1}$ be two lines.

Let $L_{3}$ be a line passing through the point $(α, β, γ)$ …
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The square of the distance of the point $(\frac{15}{7}, \frac{32}{7}, 7)$ from the line $\frac{x + 1}{3} = \frac{y + 3}{5} = \frac{z + 5}{7}$ in the direction of the vector $\hat{i} + 4 \hat{j} + 7 \hat{k}$ is :
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Let $A (x, y, z)$ be a point in xy -plane, which is equidistant from three points $(0, 3, 2), (2, 0, 3)$ and $(0, 0, 1)$ .

Let $B = (1, 4, - 1)$ and $C = (2, 0, - 2)$ . Then among the statements
(S1) : $△ ABC$ is an isosceles ri …
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If the image of the point $(4, 4, 3)$ in the line $\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - 1}{3}$ is $(α, β, γ)$ , then $α + β + γ$ is equal to
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Let P be the image of the point $Q (7, - 2, 5)$ in the line $L : \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z}{4}$ and $R (5, p, q)$ be a point on $L$ . Then the square of the area of $△ P QR$ is ____
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Let in a $△ ABC$ , the length of the side AC be 6 , the vertex B be ( $1, 2, 3$ ) and the vertices $A, C$ lie on the line $\frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{- 2}$ . Then the area (in sq. units) of …
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Let the line passing through the points $(- 1, 2, 1)$ and parallel to the line $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z}{4}$ intersect the line $\frac{x + 2}{3} = \frac{y - 3}{2} = \frac{z - 4}{1}$ at the point $P$ . Then the distance of P from the point $Q (4, - 5, 1)$ is :
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If the image of the point $P (1, 0, 3)$ in the line joining the points $A (4, 7, 1)$ and $B (3, 5, 3)$ is $Q (α, β, γ)$ , then $α + β + γ$ is equal to
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