List of top Mathematics Questions on 3D Geometry

Let P be the foot of the perpendicular from the point (1,2,2) (1, 2, 2) on the line x−11=y+1−1=z−22 \frac{x-1}{1} = \frac{y + 1}{-1} = \frac{z - 2}{2} Let the line r=(−i^+j^−2k^)+λ(i^−j^+k^) \mathbf{r} = (-\hat{i} + \hat{j} - 2\hat{k}) + \lambda (\hat{i} - \hat{j} + \hat{k}), λ∈R \lambda \in \mathbb{R} , intersect the line LL at QQ. Then 2(PQ)2 2(PQ)^2 is equal to:
The square of the distance of the point (157,327,7)\left( \frac{15}{7}, \frac{32}{7}, 7 \right) from the line x+13=y+35=z+57\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7} in the direction of the vector i+4j+7k\mathbf{i} + 4\mathbf{j} + 7\mathbf{k} is:
Let in a △ABC \triangle ABC , the length of the side AC is 6, the vertex B is (1,2,3) (1, 2, 3) and the vertices A, C lie on the line x−63=y−72=z−7−2. \frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{-2}. Then the area (in sq. units) of △ABC \triangle ABC is:
Let L1:x−13=y−1−1=z+10 L_1 : \frac{x - 1}{3} = \frac{y - 1}{-1} = \frac{z + 1}{0} and L2:x−22=y0=z+4α L_2 : \frac{x - 2}{2} = \frac{y}{0} = \frac{z + 4}{\alpha} , where α∈R \alpha \in \mathbb{R} , be two lines which intersect at the point B B . If P P is the foot of the perpendicular from the point A(1,1,−1) A(1, 1, -1) on L2 L_2 , then the value of 26α(PB)2 26 \alpha (PB)^2 is:
Let a straight line L L pass through the point P(2,−1,3)P(2, -1, 3) and be perpendicular to the lines x−12=y+11=z−3−2andx−31=y−23=z+24. \frac{x - 1}{2} = \frac{y + 1}{1} = \frac{z - 3}{-2} \quad \text{and} \quad \frac{x - 3}{1} = \frac{y - 2}{3} = \frac{z + 2}{4}. If the line LL intersects the yz-plane at the point Q, then the distance between the points P and Q is:
If the midpoint of a chord of the ellipse x29+y24=1 \frac{x^2}{9} + \frac{y^2}{4} = 1 is (2,43) \left( \sqrt{2}, \frac{4}{3} \right) , and the length of the chord is 2α3 \frac{2\sqrt{\alpha}}{3} , then α \alpha is:
Let the line x+y=1 x + y = 1 meet the circle x2+y2=4 x^2 + y^2 = 4 at the points A and B. If the line perpendicular to AB AB and passing through the mid point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ABCD is equal to:
Let ∣z1−8−2i∣≤1 |z_1 - 8 - 2i| \leq 1 and ∣z2−2+6i∣≤2 |z_2 - 2 + 6i| \leq 2 , where z1,z2∈C z_1, z_2 \in \mathbb{C} . Then the minimum value of ∣z1−z2∣ |z_1 - z_2| is:
Let A, B, C be three points in the xy-plane, whose position vectors are given by 3i^+j^ \sqrt{3} \hat{i} + \hat{j} , i^+3j^ \hat{i} + \sqrt{3} \hat{j} , and ai^+(1−a)j^ a\hat{i} + (1-a) \hat{j } respectively with respect to the origin O O . If the distance of the point C from the line bisecting the angle between the vectors OA→ \overrightarrow{OA} and OB→ \overrightarrow{OB} is 92 \frac{9}{\sqrt{2}} , then the sum of all possible values of a a is:
Two equal sides of an isosceles triangle are along −x+2y=4 -x + 2y = 4 and x+y=4 x + y = 4 . If m m is the slope of its third side, then the sum of all possible distinct values of m m is:
If A A and B B are the points of intersection of the circle x2+y2−8x=0 x^2 + y^2 - 8x = 0 and the hyperbola x29−y24=1 \frac{x^2}{9} - \frac{y^2}{4} = 1 , and a point P P moves on the line 2x−3y+4=0 2x - 3y + 4 = 0 , then the centroid of △PAB \triangle PAB lies on the line:

Let ABC ABC be a triangle formed by the lines 7x−6y+3=0 7x - 6y + 3 = 0 , x+2y−31=0 x + 2y - 31 = 0 , and 9x−2y−19=0 9x - 2y - 19 = 0 . 
Let the point (h,k) (h, k) be the image of the centroid of △ABC \triangle ABC in the line 3x+6y−53=0 3x + 6y - 53 = 0 . Then h2+k2+hk h^2 + k^2 + hk is equal to:

Let a→=i+2j+k \overrightarrow{a} = i + 2j + k and b→=2i+7j+3k \overrightarrow{b} = 2i + 7j + 3k . 
Let L1:r→=(−i+2j+k)+λa→,λ∈R L_1 : \overrightarrow{r} = (-i + 2j + k) + \lambda \overrightarrow{a}, \quad \lambda \in \mathbb{R} and L2:r→=(j+k)+μb→,μ∈R L_2 : \overrightarrow{r} = (j + k) + \mu \overrightarrow{b}, \quad \mu \in \mathbb{R} be two lines. If the line L3 L_3 passes through the point of intersection of L1 L_1 and L2 L_2 , and is parallel to a→+b→ \overrightarrow{a} + \overrightarrow{b} , then L3 L_3 passes through the point:

The angle between two lines whose direction ratios are proportional to 1,1,−21, 1, -2 and (3−1),(−3−1),−4(\sqrt{3} - 1), (-\sqrt{3} - 1), -4 is:
The direction cosines of the line which is perpendicular to the lines with direction ratios 1,-2,-2 and 0, 2, 1 are:
Consider the line L L passing through the points (1,2,3) (1, 2, 3) and (2,3,5) (2, 3, 5) . The distance of the point (113,113,193) \left( \frac{11}{3}, \frac{11}{3}, \frac{19}{3} \right) from the line L L along the line 3x−112=3y−111=3z−192 \frac{3x - 11}{2} = \frac{3y - 11}{1} = \frac{3z - 19}{2} is equal to:

If y=x−x2 y = x - x^2 , then the rate of change of y2 y^2 with respect to x2 x^2 at x=2 x = 2 is: 

The distance from a point (1,1,1) (1,1,1) to a variable plane π\pi is 12 units and the points of intersections of the plane with X, Y, Z-axes are A,B,C A, B, C respectively. If the point of intersection of the planes through the points A,B,C A, B, C and parallel to the coordinate planes is P P , then the equation of the locus of P P is: 

If A(1,0,2) A(1,0,2) , B(2,1,0) B(2,1,0) , C(2,−5,3) C(2,-5,3) , and D(0,3,2) D(0,3,2) are four points and the point of intersection of the lines AB AB and CD CD is P(a,b,c) P(a,b,c) , then a+b+c=? a + b + c = ?  

The distance of the point O(0,0,0) O(0,0,0) from the plane r→⋅(i^+j^+k^)=5 \overrightarrow{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 5 measured parallel to 2i^+3j^−6k^ 2\hat{i} + 3\hat{j} - 6\hat{k} is?
The distance of a point (2,3,−5) (2,3,-5) from the plane r⃗⋅(4i−3j+2k)=4 \vec{r} \cdot (4i - 3j + 2k) = 4 is:
If the line with direction ratios (1,a,β) (1, a, \beta) is perpendicular to the line with direction ratios (−1,2,1) (-1,2,1) and parallel to the line with direction ratios (α,1,β), (\alpha,1,\beta), then (α,β) (\alpha, \beta) is:
Let P(x1,y1,z1) P(x_1, y_1, z_1) be the foot of the perpendicular drawn from the point Q(2,−2,1) Q(2, -2, 1) to the plane x−2y+z=1. x - 2y + z = 1. If d d is the perpendicular distance from the point Q Q to the plane and I=x1+y1+z1, I = x_1 + y_1 + z_1, then I+3d2 I + 3d^2 is:
The square of the distance of the image of the point (6,1,5) (6, 1, 5) in the line x−13=y2=z−24, \frac{x - 1}{3} = \frac{y}{2} = \frac{z - 2}{4}, from the origin is _____ .
The foot of the perpendicular drawn from a point A(1,1,1) A(1,1,1) onto a plane π \pi is P(−3,3,5) P(-3,3,5) . If the equation of the plane parallel to the plane π \pi and passing through the midpoint of AP AP is ax−y+cz+d=0, ax - y + cz + d = 0, then a+c−d a + c - d is: