Three Dimensional Geometry · Mathematics · AP EAPCET

If P divides the line segment joining the points $$A(1,2,-1)$$ and $$B(-1,0,1)$$ externally in the ratio 1 : 2 and $$Q=(1,3,-1)$$, then $$PQ=$$

If the direction cosines of a line are $$\left(\frac{a}{\sqrt{83}}, \frac{5}{\sqrt{83}}, \frac{c}{\sqrt{83}}\right)$$ and $$c-a=4$$, then $$ca=$$

Let the plane $$\pi$$ pass through the point (1, 0, 1) and perpendicular to the planes $$2x + 3y - z = 2$$ and $$x - y + 2z = 1$$. Let the equation of the plane passing through the point (11, 7, 5) and parallel to the plane $$\pi$$ be $$ax + by - z - d = 0$$. Then, $${a \over b} + {b \over d} = $$

$$D, E, F$$ are respectively the points on the sides $$B C, C A$$ and $$A B$$ of a $$\triangle A B C$$ dividing them in the ratio $$2: 3,1: 2,3: 1$$ internally. The lines $$\mathbf{B E}$$ and $$\mathbf{C F}$$ intersect on the line $$\mathbf{A D}$$ at $$P$$. If $$\mathbf{A P}=x_1 \cdot \mathbf{A} \mathbf{B}+y_1 \cdot \mathbf{A C}$$, then $$x_1+y_1=$$

If the equation of the plane passing through the point $$A(-2,1,3)$$ and perpendicular to the vector $$3 \hat{i}+\hat{j}+5 \hat{k}$$ is $$a x+b y+c z+d=0$$, then $$\frac{a+b}{c+d}=$$

If $$x$$-coordinate of a point $$P$$ on the line joining the points $$Q(2,2,1)$$ and $$R(5,2,-2)$$ is 4, then the $$y$$-coordinate of $$P=$$

If $$(2,3, c)$$ are the direction ratios of a ray passing through the point $$C(5, q, 1)$$ and also the mid-point of the line segment joining the points $$A(p,-4,2)$$ and $$B(3,2,-4)$$, then $$c \cdot(p+7 q)=$$

If the equation of the plane which is at a distance of $$1 / 3$$ units from the origin and perpendicular to a line whose directional ratios are $$(1,2,2)$$ is $$x+p y+q z+r=0$$, then $$\sqrt{p^2+q^2+r^2}=$$

The point of intersection of the lines $$\mathbf{r}=2 \mathbf{b}+t(6 \mathbf{c}-\mathbf{a})$$ and $$\mathbf{r}=\mathbf{a}+s(\mathbf{b}-3 \mathbf{c})$$ is

If the point $$(a, 8,-2)$$ divides the line segment joining the points $$(1,4,6)$$ and $$(5,2,10)$$ in the ratio $$m: n$$, then $$\frac{2 m}{n}-\frac{a}{3}=$$

If $$(a, b, c)$$ are the direction ratios of a line joining the points $$(4,3,-5)$$ and $$(-2,1,-8)$$, then the point $$P(a, 3 b, 2 c)$$ lies on the plane

The $$x$$-intercept of a plane $$\pi$$ passing through the point $$(1,1,1)$$ is $$\frac{5}{2}$$ and the perpendicular distance from the origin to the plane $$\pi$$ is $$\frac{5}{7}$$. If the $$y$$-intercept of the plane $$\pi$$ is negative and the $$z$$-intercept is positive, then its $$y$$-intercept is

If the lines, $$\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{\lambda}$$ and $$\frac{x-2}{3}=\frac{y-3}{2}=\frac{z-2}{3}$$ are coplanar, then $$\sin ^{-1}(\sin \lambda)+\cos ^{-1}(\cos \lambda)$$ is equal to

The line passing through $$(1,1,-1)$$ and parallel to the vector $$\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$$ meets the line $$\frac{x-3}{-1}=\frac{y+2}{5}=\frac{z-2}{-4}$$ at $$A$$ and the plane $$2 x-y+2 z+7=0$$ at $$B$$. Then $$A B$$ is equal to

If the vertices of the triangles are (1, 2, 3), (2, 3, 1), (3, 1, 2) and if H, G, S and I respectively denote its orthocentre, centroid, circumcentre and incentre, then H + G + S + I is equal to

A(2, 3, 4), B(4, 5, 7), C(2, $$-$$6, 3) and D(4, $$-$$4, k) are four points. If the line AB is parallel to CD, then k is equal to

If the direction cosines of two lines are $$\left( {{2 \over 3},{2 \over 3},{1 \over 3}} \right)$$ and $$\left( {{5 \over {13}},{{12} \over {13}},0} \right)$$, then identify the direction ratios of a line which is bisecting one o the angle between them.

$$X$$ intercept of the plane containing the line of intersection of the planes $$x-2 y+z+2=0$$ and $$3 x-y-z+1=0$$ and also passing through $$(1,1,1)$$ is

Let $$L_1$$ (resp, $$L_2$$ ) be the line passing through $$2 \hat{\mathbf{i}}-\hat{\mathbf{k}}$$ (resp. $$2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}})$$ and parallel to $$3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$$ ( resp. $$\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$$ ). Then the shortest distance between the lines $$L_1$$ and $$L_2$$ is equal to

If the points (2, 4, $$-$$1), (3, 6, $$-$$1) and (4, 5, $$-$$1) are three consecutive vertices of a parallelogram, then its fourth vertex is

$$A(-1,2-3), B(5,0,-6)$$ and $$C(0,4,-1)$$ are the vertices of a $$\triangle A B C$$. The direction cosines of internal bisector of $$\angle B A C$$ are

If the projections of the line segment AB on xy, yz and zx planes are $$\sqrt{15},\sqrt{46},7$$ respectively, then the projection of AB on Y-axis is

Find the equation of the plane passing through the point $$(2,1,3)$$ and perpendicular to the planes $$x-2 y+2 z+3=0$$ and $$3 x-2 y+4 z-4=0$$.

The ratio in which the YZ-plane divides the line joining (2, 4, 5) and (3, 5, $$-$$4) is

The direction cosines of a line which makes equal angles with the coordinate axes are

Let $$O$$ be the origin and $$P$$ be a point which is at a distance of 3 units from the origin. If the direction ratios of $$\overline{O P}$$ are $$(1,-2,-2)$$, then the coordinates of $$P$$ are