Three Dimensional Geometry · Mathematics · AP EAPCET

If $\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}, \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}$ are the vertices of a tetrahedron, then its volume is

If a line $L$ makes angles $\frac{\pi}{3}$ and $\frac{\pi}{4}$ with $Y$-axis and $Z$-axis respectively, then the angle between $L$ and another line having direction ratio $1,1,1$ is

If $l, m$ and $n$ are the direction cosines of a line that is perpendicular to the lines having the direction ratios $(1,2,-1)$ and $(1,-2,1)$, then $(l+m+n)^2$ is equal to

The foot of the perpendicular drawn from a point $A(1,1,1)$ on to a plane $\pi$ is $P(-3,3,5)$.If the equation of the plane parallel to the plane of $\pi$ and passing through the mid-point of $A P$ is $a x-y+c z+d=0$, then $a+c-d$ is equal to

The distance of a point $(2,3,-5)$ from the plane $\hat{\mathbf{r}} \cdot(4 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+2 \hat{\mathbf{k}})=4$ is

The orthocentre of triangle fromed by points $(2,1,5)$ $(3,2,3)$ and $(4,0,4)$ is

If $P=(0,1,2), Q=(4,-2,1)$, and $O=(0,0,0)$, then $\angle P O Q=$

If the perpendicular distance from $(1,2,4)$ to the plane $2 x+2 y-z+k=0$ is 3 , then $k=$

Angle between the planes, $\mathbf{r} \cdot(12 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-3 \hat{\mathbf{k}})=5$ and, $\mathbf{r} \cdot(5 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}})=7$ is

The shortest distance between the skew lines $\mathbf{r}=(2 \hat{\mathbf{i}}-\hat{\mathbf{j}})+t(\hat{\mathbf{i}}+2 \hat{\mathbf{k}})$ and $\mathbf{r}=(-2 \hat{\mathbf{i}}+\hat{\mathbf{k}})+s(\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}})$ is

If the plane $x-y+z+4=0$ divides the line joining the points $P(2,3,-1)$ and $Q(1,4,-2)$ in the ratio $l: m$, then $l+m$ is

If the line with direction ratios $(1, \alpha, \beta)$ is perpendicular to the line with direction ratios $(-1,2,1)$ and parallel to the line with direction ratios $(\alpha, 1, \beta)$ then $(\alpha, \beta)$ is

Let $P\left(x_1, y_1, z_1\right)$ be the foot of perpendicular drawn from the point $Q(2,-2,1)$ to the plane $x-2 y+z=1$. If $d$ is the perpendicular from the point $Q$ to the plane and $l=x_1+y_1+z_1$, then $l+3 d^2$ is

$A(1,2,1), B(2,3,2), C(3,1,3)$ and $D(2,1,3)$ are the vertices of a tetrahedron. If $\theta$ is the angle between the faces $A B C$ and $A B D$, then $\cos \theta=$

Consider the tetrahedron with the vertices $A(3,2,4)$, $B\left(x_1, y_1, 0\right), C\left(x_2, y_2, 0\right)$ and $D\left(x_3, y_3, 0\right)$.If the $\triangle B C D$ is formed by the lines $y=x, x+y=6$ and $y=1$, then the centroid of the tetrahedron is

If $P(2, \beta, \alpha)$ lies on the plane $x+2 y-z-2=0$ and $Q(\alpha,-1, \beta)$ lies on the plane $2 x-y+3 z+6=0$, then the direction cosines of the $P Q$ are

Let $\pi$ be the plane that passes through the point $(-2,1,-1)$ and parallel to the plane $2 x-y+2 z=0$. Then the foot of perpendicular drawn from the point $(1,2,1)$ to the plane $\pi$ is

The angle between the line with the direction ratios $(2,5,1)$ and the plane $8 x+2 y-z=14$ is

AP EAPCET 2024 - 20th May Evening Shift

The direction cosines of the line of intersection of the planes $x+2 y+z-4=0$ and $2 x-y+z-3=0$ are

AP EAPCET 2024 - 20th May Evening Shift

If $L_1$ and $L_2$ are two lines which pass through origin and having direction ratios $(3,1,-5)$ and $(2,3,-1)$ respectively, then equation of the plane containing $L_1$ and $L_2$ is

AP EAPCET 2024 - 20th May Evening Shift

Let $O(\mathbf{O}), A(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}), B(-2 \hat{\mathbf{i}}+3 \hat{\mathbf{k}}), C(2 \hat{\mathbf{i}}+\hat{\mathbf{j}})$ and $D(4 \hat{\mathbf{k}})$ are position vectors of the points $O, A, B, C$ and $D$. If a line passing through $A$ and $B$ intersects the plane passing through $O, C$ and $D$ at the point $R$, then position vector of $R$ is

The distance of the point $O(\mathbf{O})$ from the plane $\mathbf{r}$. $(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})=5$ measured parallel to $2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-6 \hat{\mathbf{k}}$ is

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

The direction cosines of two lines are connected by the relations $l+m-n=0$ and $l m-2 m n+n l=0$. If $\theta$ is the acute angle between those lines, then $\cos \theta=$

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

The shortest distance between the skew lines $\mathbf{r}=(-\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}})+t(3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}})$ and $\mathbf{r}=(7 \hat{\mathbf{i}}+4 \hat{\mathbf{k}})+s(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}})$ is

If $A(1,2,0), B(2,0,1), C(-3,0,2)$ are the vertices of $\triangle A B C$, then the length of the internal bisector of $\angle B A C$ is

The perpendicular distance from the point $(-1,1,0)$ to the line joining the points $(0,2,4)$ and $(3,0,1)$ is

A line $L$ passes through the points $(1,2,-3)$ and $(\beta, 3,1)$ and a plane $\pi$ passes through the points $(2,1,-2)$, $(-2,-3,6),(0,2,-1)$. If $\theta$ is the angle between the line $L$ and plane $\pi$, then $27 \cos ^2 \theta=$

If the points with position vectors $(\alpha \hat{\mathbf{i}}+10 \hat{\mathbf{j}}+13 \hat{\mathbf{k}}),(6 \hat{\mathbf{i}}+11 \hat{\mathbf{j}}+11 \hat{\mathbf{k}}),\left(\frac{9}{2} \hat{\mathbf{i}}+\beta \hat{\mathbf{j}}-8 \hat{\mathbf{k}}\right)$ are collinear, then $(19 \alpha-6 \beta)^2=$

The equation $a x y+b y=c y$ represents the locus of the points which lie on

Let $P(\alpha, 4,7)$ and $Q(\beta, \beta, 8)$ be two points. If $Y Z$-plane divides the join of the points $P$ and $Q$ in the ratio $2: 3$ and $Z X$-plane divides the join of $P$ and $Q$ in the ratio $4: 5$, then length of line segment $P Q$ is

If the distance between the planes $2 x+y+z+1=0$ and $2 x+y+z+\alpha=0$ is 3 units, then product of all possible values of $\alpha$ is

AP EAPCET 2022 - 5th July Morning Shift

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 $$ca=$$

AP EAPCET 2022 - 5th July Morning Shift

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, $${a \over b} + {b \over d} = $$

AP EAPCET 2022 - 5th July Morning Shift

$$D, E, F$$ 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 $$3 x-2 y+4 z-4=0$$.