Three-Dimensional Geometry

If the lines $\vec r=\left(\hat i-\hat j+\hat k\right)+\lambda\left(3\hat j-\hat k\right)$ and $\vec r=\left(\alpha\hat i-\hat j\right)+\mu\left(2\hat i-3\hat k\right)$ are co-planar, then the distance of the plane containing these two lines from the point $(\alpha,0,0)$ is:

If two straight lines whose direction cosines are given by the relations $l+m-n=0$, $3l^2+m^2+cnl=0$ are parallel, then the positive value of $c$ is :

Let the mirror image of the point $(a,b,c)$ with respect to the plane $3x-4y+12z+19=0$ be $(a-6,\beta,\gamma)$. If $a+b+c=5$, then $7\beta-9\gamma$ is equal to ______.

Let the foot of the perpendicular from the point $(1,2,4)$ on the line $\dfrac{x+2}4=\dfrac{y-1}2=\dfrac{z+1}3$ be $P$. Then the distance of $P$ from the plane $3x+4y+12z+23=0$ is

The shortest distance between the lines $\dfrac{x-3}2=\dfrac{y-2}3=\dfrac{z-1}{-1}$ and $\dfrac{x+3}2=\dfrac{y-6}1=\dfrac{z-5}3$, is

If two distinct points Q, R lie on the line of intersection of the planes $-x+2y-z=0$ and $3x-5y+2z=0$ and $PQ=PR=\sqrt{18}$ where the point P is $(1,-2,3)$, then the area of the triangle PQR is equal to

The acute angle between the planes $P_1$ and $P_2$, when $P_1$ and $P_2$ are the planes passing through the intersection of the planes $5x+8y+13z-29=0$ and $8x-7y+z-20=0$ and the points $(2,1,3)$ and $(0,1,2)$, respectively, is

Let the plane $P:\vec r\cdot\vec a=d$ contain the line of intersection of two planes $\vec r\cdot\left(\hat i+3\hat j-\hat k\right)=6$ and $\vec r\cdot\left(-6\hat i+5\hat j-\hat k\right)=7$. If the plane P passes through the point $\left(2,3,\dfrac12\right)$, then the value of $\dfrac{\left|13\vec a\right|^2}{d^2}$ is equal to

Let the plane $ax+by+cz=d$ pass through $(2,3,-5)$ and is perpendicular to the planes $2x+y-5z=10$ and $3x+5y-7z=12$.

If $a,b,c,d$ are integers $d>0$ and $\gcd(|a|,|b|,|c|,d)=1$, then the value of $a+7b+c+20d$ is equal to :

Let the image of the point $P(1,2,3)$ in the line $L:\dfrac{x-6}3=\dfrac{y-1}2=\dfrac{z-2}3$ be Q. Let $R(\alpha,\beta,\gamma)$ be a point that divides internally the line segment PQ in the ratio $1:3$. Then the value of $22(\alpha+\beta+\gamma)$ is equal to ______.

If the mirror image of the point $(2,4,7)$ in the plane $3x-y+4z=2$ is $(a,b,c)$, then $2a+b+2c$ is equal to :

Let d be the distance between the foot of perpendiculars of the points $P(1,2,-1)$ and $Q(2,-1,3)$ on the plane $-x+y+z=1$. Then $d^2$ is equal to ______.

Let $P_1:\vec r\cdot\left(2\hat i+\hat j-3\hat k\right)=4$ be a plane. Let $P_2$ be another plane which passes through the points $(2,-3,2)$, $(2,-2,-3)$ and $(1,-4,2)$. If the direction ratios of the line of intersection of $P_1$ and $P_2$ be $16,\alpha,\beta$, then the value of $\alpha+\beta$ is equal to ______.

Let $\dfrac{x-2}3=\dfrac{y+1}{-2}=\dfrac{z+3}{-1}$ lie on the plane $px-qy+z=5$, for some $p,q\in\mathbb R$. The shortest distance of the plane from the origin is:

Let Q be the mirror image of the point $P(1,2,1)$ with respect to the plane $x+2y+2z=16$. Let T be a plane passing through the point Q and contains the line $\vec r=-\hat k+\lambda\left(\hat i+\hat j+2\hat k\right)$, $\lambda\in\mathbb R$. Then, which of the following points lies on T?

The distance of the point $(3,2,-1)$ from the plane $3x-y+4z+1=0$ along the line $\dfrac{2-x}2=\dfrac{y-3}2=\dfrac{z+1}1$ is equal to :

Consider a triangle ABC whose vertices are $A(0,\alpha,\alpha)$, $B(\alpha,0,\alpha)$ and $C(\alpha,\alpha,0)$, $\alpha>0$. Let D be a point moving on the line $x+z-3=0=y$ and G be the centroid of $\Delta ABC$. If the minimum length of GD is $\sqrt{\dfrac{57}2}$, then $\alpha$ is equal to ______.

Let P be the plane containing the straight line $\dfrac{x-3}9=\dfrac{y+4}{-1}=\dfrac{z-7}{-5}$ and perpendicular to the plane containing the straight lines $\dfrac x2=\dfrac y3=\dfrac z5$ and $\dfrac x3=\dfrac y7=\dfrac z8$. If d is the distance of P from the point $(2,-5,11)$, then $d^2$ is equal to :

The line of shortest distance between the lines $\dfrac{x-2}0=\dfrac{y-1}1=\dfrac z1$ and $\dfrac{x-3}2=\dfrac{y-5}2=\dfrac{z-1}1$ makes an angle of $\cos^{-1}\left(\sqrt{\dfrac2{27}}\right)$ with the plane $P:ax-y-z=0$, $(a>0)$. If the image of the point $(1,1,-5)$ in the plane P is $(\alpha,\beta,\gamma)$, then $\alpha+\beta-\gamma$ is equal to ______.

A plane E is perpendicular to the two planes $2x-2y+z=0$ and $x-y+2z=4$, and passes through the point $P(1,-1,1)$. If the distance of the plane E from the point $Q(a,a,2)$ is $3\sqrt2$, then $(PQ)^2$ is equal to