3D Geometry
Let the line $L: \frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-3}{1}$ intersect the plane $2 x+y+3 z=16$ at the point $P$. Let the point $Q$ be the foot of perpendicular from the point $R(1,-1,-3)$ on the line $L$. If $\alpha$ is the area of triangle $P Q R$, then $\alpha^{2}$ is equal to __________.
Explanation:
Let $P \equiv\left(2 r_{1}+1,-r_{1}, r_{1}+3\right)$
$P$ lies on $2 x+y+3 z=16$
$\therefore 2\left(2 r_{1}+1\right)+\left(-r_{1}-1\right)+3\left(r_{1}+3\right)=16$
$r_{1}=1$
$P \equiv(3,-2,4)$
$R \equiv(1,-1,-3)$
Let $Q \equiv\left(2 r_{2}+1,-r_{2}-1, r_{2}+3\right)$
$D R$ s of $Q R \equiv\left(2 r_{2}-r_{2} r_{2}+6\right)$
DRs of $L \equiv(2,-1,1)$
$Q R \perp L \Rightarrow 4 r_{2}+r_{2}+r_{2}+6=0$
$r_{2}=-1$
$Q \equiv(-1,0,2)$
$\overrightarrow{Q P} \times \overrightarrow{R P}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 4 & -2 & 2 \\ 2 & -1 & 7\end{array}\right|=-12 \hat{i}-24 \hat{j}+0 \hat{k}$
$\alpha=[P Q R]=\frac{1}{2}|\overrightarrow{Q P} \times \overrightarrow{R P}|=\frac{1}{2} \times 12 \sqrt{5}$
$=6 \sqrt{5}$
$\alpha^{2}=180$
Let $\theta$ be the angle between the planes $P_{1}: \vec{r} \cdot(\hat{i}+\hat{j}+2 \hat{k})=9$ and $P_{2}: \vec{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})=15$. Let $\mathrm{L}$ be the line that meets $P_{2}$ at the point $(4,-2,5)$ and makes an angle $\theta$ with the normal of $P_{2}$. If $\alpha$ is the angle between $\mathrm{L}$ and $P_{2}$, then $\left(\tan ^{2} \theta\right)\left(\cot ^{2} \alpha\right)$ is equal to ____________.
Explanation:
$ \begin{aligned} & P_{2}: \vec{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})=15 \\\\ & \text { then } \cos \theta=\frac{3}{\sqrt{6} \cdot \sqrt{6}}=\frac{1}{2} \\\\ & \begin{aligned} & \therefore \theta=\frac{\pi}{3}, \text { Now } \alpha=\frac{\pi}{2}-\theta \\\\ & \therefore \tan ^{2} \theta \cdot \cot ^{2} \alpha=\tan ^{4} \theta \\\\ &=(\sqrt{3})^{4}=9 \end{aligned} \end{aligned} $
Explanation:
$L:{{x - 2} \over 1} = {{y - 3} \over { - 1}} = {{z - 1} \over { - 1}} = \lambda $
Any point on L can be taken as
$B(\lambda + 2, - \lambda + 3, - \lambda + 1)$
Let $A(5,3,8)$
So, $AB\,.\,(\widehat i - \widehat j - \widehat k) = 0$
$[(\lambda - 3)\widehat i - \lambda \widehat j - (\lambda + 7)\widehat k]\,.\,[\widehat i - \widehat j - \widehat k] = 0$
$\lambda - 3 + \lambda + \lambda + 7 = 0$
$\therefore$ $\lambda = {{ - 4} \over 3}$
$\overrightarrow {AB} = {{13} \over 3}\widehat i + {4 \over 3}\widehat i - {{17} \over 3}\widehat k$
$|\overrightarrow {AB} | = \sqrt {{{169} \over 9} + {{16} \over 9} + {{289} \over 9}} $
$ = {{\sqrt {474} } \over 3} = \alpha $
$3{\alpha ^2} = {{474} \over 9} \times 3 = 158$
If the equation of the plane passing through the point $(1,1,2)$ and perpendicular to the line $x-3 y+ 2 z-1=0=4 x-y+z$ is $\mathrm{A} x+\mathrm{B} y+\mathrm{C} z=1$, then $140(\mathrm{C}-\mathrm{B}+\mathrm{A})$ is equal to ___________.
Explanation:
Line of intersection of the planes $x - 3y + 2z - 1 = 0$ and $4x - y + z = 0$ is normal $(\overrightarrow n )$ to the required plane.
$\overrightarrow n = \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 1 & { - 3} & 2 \cr 4 & { - 1} & 1 \cr } } \right| = - \widehat i + 7\widehat j + 11\widehat k$
Equation of plane is
$ - x + 7y + 11z = \lambda $
It passes through (1, 1, 2)
$\therefore$ $\lambda = 28$
So, the plane is
$ - x + 7y + 11z = 28$
$ \Rightarrow {{ - 1} \over {28}}x + {7 \over {28}}y + {{11} \over {28}}z = 1$
$A = {{ - 1} \over {28}},B = {7 \over {28}},C = {{11} \over {28}}$
$140(C - B + A) = 15$
If $\lambda_{1} < \lambda_{2}$ are two values of $\lambda$ such that the angle between the planes $P_{1}: \vec{r}(3 \hat{i}-5 \hat{j}+\hat{k})=7$ and $P_{2}: \vec{r} \cdot(\lambda \hat{i}+\hat{j}-3 \hat{k})=9$ is $\sin ^{-1}\left(\frac{2 \sqrt{6}}{5}\right)$, then the square of the length of perpendicular from the point $\left(38 \lambda_{1}, 10 \lambda_{2}, 2\right)$ to the plane $P_{1}$ is ______________.
Explanation:
${P_1}:\overrightarrow r \,.\,(3\widehat i - 5\widehat j + \widehat k) = 7$
${P_2}:\overrightarrow r \,.\,(\lambda \widehat i + \widehat j - 3\widehat k) = 9$
Let angle between P1 and P2 is $\theta$
Then $\cos \theta = {{3\lambda - 5 - 3} \over {\sqrt {35} \sqrt {{\lambda ^2} + 10} }}$
But $\sin \theta = {{2\sqrt 6 } \over 5}$
$\therefore$ ${{{{(3\lambda - 8)}^2}} \over {35({\lambda ^2} + 10)}} = 1 - {{24} \over {25}}$
$ \Rightarrow 5(9{\lambda ^2} + 64 - 48\lambda ) = 7{\lambda ^2} + 70$
$ \Rightarrow 38{\lambda ^2} - 240\lambda + 250 = 0$
$ \Rightarrow 19{\lambda ^2} - 120\lambda + 125 = 0$
$ \Rightarrow (19\lambda - 25)(\lambda - 5) = 0$
$\therefore$ ${\lambda _1} = {{25} \over {19}},{\lambda _2} = 5$
So, point (50, 50, 2)
$\therefore$ $d = {{|150 - 250 + 2 - 7|} \over {\sqrt {35} }} = 315$
Let the equation of the plane P containing the line $x+10=\frac{8-y}{2}=z$ be $ax+by+3z=2(a+b)$ and the distance of the plane $P$ from the point (1, 27, 7) be $c$. Then $a^2+b^2+c^2$ is equal to __________.
Explanation:
$\because$ the plane $a x+b y+3 z=2(a+b)$
$\Rightarrow \mathrm{b}=2 \mathrm{a}$
$\&$ dot product of d.r.'s is zero
$\therefore \mathrm{a}-2 \mathrm{~b}+3=0$ $\therefore \mathrm{a}=1 \& \mathrm{~b}=2$
Distance from $(1,27,7)$ is
$c=\frac{1+54+21-6}{\sqrt{14}}=\frac{70}{\sqrt{14}}=5 \sqrt{14}$
$\therefore \mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}=1+4+350$
$=355$
Let the co-ordinates of one vertex of $\Delta ABC$ be $A(0,2,\alpha)$ and the other two vertices lie on the line ${{x + \alpha } \over 5} = {{y - 1} \over 2} = {{z + 4} \over 3}$. For $\alpha \in \mathbb{Z}$, if the area of $\Delta ABC$ is 21 sq. units and the line segment $BC$ has length $2\sqrt{21}$ units, then $\alpha^2$ is equal to ___________.
Explanation:
A. $\left(\mathrm{O}_{1} 2, \alpha\right)$
$\sqrt{(2 \alpha+5)^{2}+(2 \alpha+20)^{2}+(2 \alpha-5)^{2}}=\sqrt{21} \sqrt{38}$
$\Rightarrow 12 \alpha^{2}+80 \alpha+450=798$
$\Rightarrow 12 \alpha^{2}+80 \alpha-348=0$
$\Rightarrow \alpha=3 \Rightarrow \alpha^{2}=9$
If the shortest distance between the line joining the points (1, 2, 3) and (2, 3, 4), and the line ${{x - 1} \over 2} = {{y + 1} \over { - 1}} = {{z - 2} \over 0}$ is $\alpha$, then 28$\alpha^2$ is equal to ____________.
Explanation:
$L_{1}: \frac{(x-1)}{1}=\frac{(y-2)}{1}=\frac{(2-3)}{1}$
$L_{2}: \frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-2}{0}$
$\vec{b}_{1}=\hat{i}+\hat{j}+\hat{k}$
$\vec{b}_{2}=2 \hat{i}-\hat{j}+0 \hat{k}$
$ \begin{aligned} & \overrightarrow{a_{1}}-\overrightarrow{a_{2}}=0 \hat{i}-3 \hat{j}-\hat{k} \\\\ & d=\left|\frac{\left(\bar{a}_{1}-\bar{a}_{2}\right) \cdot\left(n_{1} \times n_{2}\right)}{\left|n_{1} \times n_{2}\right|}\right| \\\\ &=\left|\frac{6-3}{\sqrt{9+1+4}}\right|=\frac{3}{\sqrt{14}}=\alpha \\\\ & 28 \alpha^{2}=\frac{28 \times 9}{14}=18 \end{aligned} $
Let the equation of the plane passing through the line $x - 2y - z - 5 = 0 = x + y + 3z - 5$ and parallel to the line $x + y + 2z - 7 = 0 = 2x + 3y + z - 2$ be $ax + by + cz = 65$. Then the distance of the point (a, b, c) from the plane $2x + 2y - z + 16 = 0$ is ____________.
Explanation:
$(x-2 y-z-5)+\lambda(x+y+3 z-5)=0$
$\because \quad$ it's parallel to the line
$x+y+2 z-7=0=2 x+3 y+z-2$
So, vector along the line $\left|\begin{array}{lll}\hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & 2 \\ 2 & 3 & 1\end{array}\right|$
$ =-5 \hat{i}+3 \hat{j}+\hat{k} $
$\because $ Plane is parallel to line
$\therefore -5(1+\lambda)+3(-2+\lambda)+1(-1+3 \lambda)=0$
$ \lambda=12 $
So, by (i)
$13 x+10 y+35 z=65$
$\therefore a=13, b=10, c=35$ and $d=\frac{26+20-35+16}{\sqrt{9}}=9$
If the shortest between the lines ${{x + \sqrt 6 } \over 2} = {{y - \sqrt 6 } \over 3} = {{z - \sqrt 6 } \over 4}$ and ${{x - \lambda } \over 3} = {{y - 2\sqrt 6 } \over 4} = {{z + 2\sqrt 6 } \over 5}$ is 6, then the square of sum of all possible values of $\lambda$ is :
Explanation:
$\frac{x+\sqrt{6}}{2}=\frac{y-\sqrt{6}}{3}=\frac{z-\sqrt{6}}{4}$
$\frac{x-\lambda}{3}=\frac{y-2 \sqrt{6}}{4}=\frac{2+2 \sqrt{6}}{5}$ is 6
Vector along line of shortest distance
$=\left|\begin{array}{lll}\mathrm{i} & \mathrm{j} & \mathrm{k} \\ 2 & 3 & 4 \\ 3 & 4 & 5\end{array}\right|, \Rightarrow-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\mathrm{k}$ (its magnitude is $\sqrt{6}$ )
Now $\frac{1}{\sqrt{6}}\left|\begin{array}{ccc}\sqrt{6}+\lambda & \sqrt{6} & -3 \sqrt{6} \\ 2 & 3 & 4 \\ 3 & 4 & 5\end{array}\right|=\pm 6$
$\Rightarrow \lambda=-2 \sqrt{6}, 10 \sqrt{6}$
So, square of sum of these values
$ =(10 \sqrt{6}-2 \sqrt{6})^2=(8 \sqrt{6})^2=384 $
The shortest distance between the lines ${{x - 2} \over 3} = {{y + 1} \over 2} = {{z - 6} \over 2}$ and ${{x - 6} \over 3} = {{1 - y} \over 2} = {{z + 8} \over 0}$ is equal to ________
Explanation:
For ${L_1}:$${{x - 2} \over 3} = {{y + 1} \over 2} = {{z - 6} \over 2}$
$\therefore$ Point on line is $A(2,-1,6)$
Parallel vector to this line is,
$\overrightarrow p = 3\widehat i + 2\widehat j + 2\widehat k$
For ${L_2}:$${{x - 6} \over 3} = {{y - 1} \over -2} = {{z + 8} \over 0}$
$\therefore$ Point on line is $B(6,1,-8)$
Parallel vector to this line is,
$\overrightarrow q = 3\widehat i - 2\widehat j + 0\widehat k$
Now shortest distance between this two lines is,
$ = \left| {{{\overrightarrow {AB} \,.\,\left( {\overrightarrow p \times \overrightarrow q } \right)} \over {\left| {\overrightarrow p \times \overrightarrow q } \right|}}} \right|$
Now, $\overrightarrow {AB} = 4 \widehat i + 2\widehat j -14\widehat k$
$\overrightarrow p \times \overrightarrow q = \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 3 & 2 & 2 \cr 3 & -2 & 0 \cr } } \right|$
$ = 4\widehat i +6\widehat j -12\widehat k$
$\therefore$ $|\vec{p} \times \vec{q}|=\sqrt{16+36+144}=\sqrt{196}=14$
and $\overrightarrow {AB} \,.\,\left( {\overrightarrow p \times \overrightarrow q } \right) = + 16 + 12 + 168 = +196$
$\therefore$ Shortest distance $ = \left| {{{ 196} \over {14 }}} \right| = 14$
Let $Q$ be the foot of perpendicular drawn from the point $P(1,2,3)$ to the plane $x+2 y+z=14$. If $R$ is a point on the plane such that $\angle P R Q=60^{\circ}$, then the area of $\triangle P Q R$ is equal to :
If $(2,3,9),(5,2,1),(1, \lambda, 8)$ and $(\lambda, 2,3)$ are coplanar, then the product of all possible values of $\lambda$ is:
If the foot of the perpendicular from the point $\mathrm{A}(-1,4,3)$ on the plane $\mathrm{P}: 2 x+\mathrm{m} y+\mathrm{n} z=4$, is $\left(-2, \frac{7}{2}, \frac{3}{2}\right)$, then the distance of the point A from the plane P, measured parallel to a line with direction ratios $3,-1,-4$, is equal to :
Let the lines
$\frac{x-1}{\lambda}=\frac{y-2}{1}=\frac{z-3}{2}$ and
$\frac{x+26}{-2}=\frac{y+18}{3}=\frac{z+28}{\lambda}$ be coplanar
and $\mathrm{P}$ be the plane containing these two lines.
Then which of the following points does NOT lie on P?
A plane P is parallel to two lines whose direction ratios are $-2,1,-3$ and $-1,2,-2$ and it contains the point $(2,2,-2)$. Let P intersect the co-ordinate axes at the points $\mathrm{A}, \mathrm{B}, \mathrm{C}$ making the intercepts $\alpha, \beta, \gamma$. If $\mathrm{V}$ is the volume of the tetrahedron $\mathrm{OABC}$, where $\mathrm{O}$ is the origin, and $\mathrm{p}=\alpha+\beta+\gamma$, then the ordered pair $(\mathrm{V}, \mathrm{p})$ is equal to :
The foot of the perpendicular from a point on the circle $x^{2}+y^{2}=1, z=0$ to the plane $2 x+3 y+z=6$ lies on which one of the following curves?
If the length of the perpendicular drawn from the point $P(a, 4,2)$, a $>0$ on the line $\frac{x+1}{2}=\frac{y-3}{3}=\frac{z-1}{-1}$ is $2 \sqrt{6}$ units and $Q\left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right)$ is the image of the point P in this line, then $\mathrm{a}+\sum\limits_{i=1}^{3} \alpha_{i}$ is equal to :
If the line of intersection of the planes $a x+b y=3$ and $a x+b y+c z=0$, a $>0$ makes an angle $30^{\circ}$ with the plane $y-z+2=0$, then the direction cosines of the line are :
If the plane $P$ passes through the intersection of two mutually perpendicular planes $2 x+k y-5 z=1$ and $3 k x-k y+z=5, k<3$ and intercepts a unit length on positive $x$-axis, then the intercept made by the plane $P$ on the $y$-axis is :
A vector $\vec{a}$ is parallel to the line of intersection of the plane determined by the vectors $\hat{i}, \hat{i}+\hat{j}$ and the plane determined by the vectors $\hat{i}-\hat{j}, \hat{i}+\hat{k}$. The obtuse angle between $\vec{a}$ and the vector $\vec{b}=\hat{i}-2 \hat{j}+2 \hat{k}$ is :
The length of the perpendicular from the point $(1,-2,5)$ on the line passing through $(1,2,4)$ and parallel to the line $x+y-z=0=x-2 y+3 z-5$ is :
A plane $E$ is perpendicular to the two planes $2 x-2 y+z=0$ and $x-y+2 z=4$, and passes through the point $P(1,-1,1)$. If the distance of the plane $E$ from the point $Q(a, a, 2)$ is $3 \sqrt{2}$, then $(P Q)^{2}$ is equal to :
The shortest distance between the lines $\frac{x+7}{-6}=\frac{y-6}{7}=z$ and $\frac{7-x}{2}=y-2=z-6$ is :
Let $\mathrm{P}$ be the plane containing the straight line $\frac{x-3}{9}=\frac{y+4}{-1}=\frac{z-7}{-5}$ and perpendicular to the plane containing the straight lines $\frac{x}{2}=\frac{y}{3}=\frac{z}{5}$ and $\frac{x}{3}=\frac{y}{7}=\frac{z}{8}$. If $\mathrm{d}$ is the distance of $\mathrm{P}$ from the point $(2,-5,11)$, then $\mathrm{d}^{2}$ is equal to :
The distance of the point (3, 2, $-$1) from the plane $3x - y + 4z + 1 = 0$ along the line ${{2 - x} \over 2} = {{y - 3} \over 2} = {{z + 1} \over 1}$ is equal to :
Let ${{x - 2} \over 3} = {{y + 1} \over { - 2}} = {{z + 3} \over { - 1}}$ lie on the plane $px - qy + z = 5$, for some p, q $\in$ 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 $\overrightarrow r = - \widehat k + \lambda \left( {\widehat i + \widehat j + 2\widehat k} \right),\,\lambda \in R$. Then, which of the following points lies on T?
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 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 :
If two distinct point 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 P1 and P2, when P1 and P2 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 $P:\overrightarrow r \,.\,\overrightarrow a = d$ contain the line of intersection of two planes $\overrightarrow r \,.\,\left( {\widehat i + 3\widehat j - \widehat k} \right) = 6$ and $\overrightarrow r \,.\,\left( { - 6\widehat i + 5\widehat j - \widehat k} \right) = 7$. If the plane P passes through the point $\left( {2,3,{1 \over 2}} \right)$, then the value of ${{|13\overrightarrow a {|^2}} \over {{d^2}}}$ is equal to :
Let the foot of the perpendicular from the point (1, 2, 4) on the line ${{x + 2} \over 4} = {{y - 1} \over 2} = {{z + 1} \over 3}$ be P. Then the distance of P from the plane $3x + 4y + 12z + 23 = 0$ is :
The shortest distance between the lines
${{x - 3} \over 2} = {{y - 2} \over 3} = {{z - 1} \over { - 1}}$ and ${{x + 3} \over 2} = {{y - 6} \over 1} = {{z - 5} \over 3}$, is :
If two straight lines whose direction cosines are given by the relations $l + m - n = 0$, $3{l^2} + {m^2} + cnl = 0$ are parallel, then the positive value of c is :
If the plane $2x + y - 5z = 0$ is rotated about its line of intersection with the plane $3x - y + 4z - 7 = 0$ by an angle of ${\pi \over 2}$, then the plane after the rotation passes through the point :
If the lines $\overrightarrow r = \left( {\widehat i - \widehat j + \widehat k} \right) + \lambda \left( {3\widehat j - \widehat k} \right)$ and $\overrightarrow r = \left( {\alpha \widehat i - \widehat j} \right) + \mu \left( {2\widehat i - 3\widehat k} \right)$ are co-planar, then the distance of the plane containing these two lines from the point ($\alpha$, 0, 0) is :
Let $\overrightarrow a = \widehat i + \widehat j + 2\widehat k$, $\overrightarrow b = 2\widehat i - 3\widehat j + \widehat k$ and $\overrightarrow c = \widehat i - \widehat j + \widehat k$ be three given vectors. Let $\overrightarrow v $ be a vector in the plane of $\overrightarrow a $ and $\overrightarrow b $ whose projection on $\overrightarrow c $ is ${2 \over {\sqrt 3 }}$. If $\overrightarrow v \,.\,\widehat j = 7$, then $\overrightarrow v \,.\,\left( {\widehat i + \widehat k} \right)$ is equal to :
If the two lines ${l_1}:{{x - 2} \over 3} = {{y + 1} \over {-2}},\,z = 2$ and ${l_2}:{{x - 1} \over 1} = {{2y + 3} \over \alpha } = {{z + 5} \over 2}$ are perpendicular, then an angle between the lines l2 and ${l_3}:{{1 - x} \over 3} = {{2y - 1} \over { - 4}} = {z \over 4}$ is :
Let the plane 2x + 3y + z + 20 = 0 be rotated through a right angle about its line of intersection with the plane x $-$ 3y + 5z = 8. If the mirror image of the point $\left( {2, - {1 \over 2},2} \right)$ in the rotated plane is B(a, b, c), then :
Let p be the plane passing through the intersection of the planes $\overrightarrow r \,.\,\left( {\widehat i + 3\widehat j - \widehat k} \right) = 5$ and $\overrightarrow r \,.\,\left( {2\widehat i - \widehat j + \widehat k} \right) = 3$, and the point (2, 1, $-$2). Let the position vectors of the points X and Y be $\widehat i - 2\widehat j + 4\widehat k$ and $5\widehat i - \widehat j + 2\widehat k$ respectively. Then the points :
Let Q be the mirror image of the point P(1, 0, 1) with respect to the plane S : x + y + z = 5. If a line L passing through (1, $-$1, $-$1), parallel to the line PQ meets the plane S at R, then QR2 is equal to :
If the shortest distance between the lines ${{x - 1} \over 2} = {{y - 2} \over 3} = {{z - 3} \over \lambda }$ and ${{x - 2} \over 1} = {{y - 4} \over 4} = {{z - 5} \over 5}$ is ${1 \over {\sqrt 3 }}$, then the sum of all possible value of $\lambda$ is :
Let the points on the plane P be equidistant from the points ($-$4, 2, 1) and (2, $-$2, 3). Then the acute angle between the plane P and the plane 2x + y + 3z = 1 is :
Let a line with direction ratios $a,-4 a,-7$ be perpendicular to the lines with direction ratios $3,-1,2 b$ and $b, a,-2$. If the point of intersection of the line $\frac{x+1}{a^{2}+b^{2}}=\frac{y-2}{a^{2}-b^{2}}=\frac{z}{1}$ and the plane $x-y+z=0$ is $(\alpha, \beta, \gamma)$, then $\alpha+\beta+\gamma$ is equal to _________.
Explanation:
Given $a\,.\,3 + ( - 4a)( - 1) + ( - 7)2b = 0$ ...... (1)
and $ab - 4{a^2} + 14 = 0$ ....... (2)
$ \Rightarrow {a^2} = 4$ and ${b^2} = 1$
$\therefore$ $L \equiv {{x + 1} \over 5} = {{y - 2} \over 3} = {z \over 1} = \lambda $ (say)
$\Rightarrow$ General point on line is $(5\lambda - 1,\,3\lambda + 2,\,\lambda )$ for finding point of intersection with $x - y + z = 0$ we get $(5\lambda - 1) - (3\lambda + 2) + (\lambda ) = 0$
$ \Rightarrow 3\lambda - 3 = 0 \Rightarrow \lambda = 1$
$\therefore$ Point at intersection $(4,\,5,\,1)$
$\therefore$ $\alpha + \beta + \gamma = 4 + 5 + 1 = 10$
Let $\mathrm{P}(-2,-1,1)$ and $\mathrm{Q}\left(\frac{56}{17}, \frac{43}{17}, \frac{111}{17}\right)$ be the vertices of the rhombus PRQS. If the direction ratios of the diagonal RS are $\alpha,-1, \beta$, where both $\alpha$ and $\beta$ are integers of minimum absolute values, then $\alpha^{2}+\beta^{2}$ is equal to ____________.
Explanation:
d.r's of $RS = < \alpha , - 1,\beta > $
d.r's of $PQ = < {{90} \over {17}},{{60} \over {17}},{{94} \over {17}} > \, = \, < 45,30,47 > $
as PQ and RS are diagonals of rhombus
$\alpha (45) + 30( - 1) + 47(\beta ) = 0$
$ \Rightarrow 45\alpha + 47\beta = 30$
i.e., $\alpha = {{30 - 47\beta } \over {45}}$
for minimum integral value $\alpha = - 15$ and $\beta = 15$
$ \Rightarrow {\alpha ^2} + {\beta ^2} = 450$.
Let the line $\frac{x-3}{7}=\frac{y-2}{-1}=\frac{z-3}{-4}$ intersect the plane containing the lines $\frac{x-4}{1}=\frac{y+1}{-2}=\frac{z}{1}$ and $4 a x-y+5 z-7 a=0=2 x-5 y-z-3, a \in \mathbb{R}$ at the point $P(\alpha, \beta, \gamma)$. Then the value of $\alpha+\beta+\gamma$ equals _____________.
Explanation:
Equation of plane containing the line $4ax - y + 5z - 7a = 0 = 2x - 5y - z - 3$ can be written as
$4ax - y + 5z - 7a + \lambda (2x - 5y - z - 3) = 0$
$(4a + 2\lambda )x - (1 + 5\lambda )y + (5 - \lambda )z - (7z + 3\lambda ) = 0$
Which is coplanar with the line
${{x - 4} \over 1} = {{y + 1} \over { - 2}} = {z \over 1}$
$4(4a + 2\lambda ) + (1 + 5\lambda ) - (7a + 3\lambda ) = 0$
$9a + 10\lambda + 1 = 0$ ..... (1)
$(4a + 2\lambda )1 + (1 + 5\lambda )2 + 5 - \lambda = 0$
$4a + 11\lambda + 7 = 0$ ...... (2)
$a = 1,\,\lambda = - 1$
Equation of plane is $x + 2y + 3z - 2 = 0$
Intersection with the line
${{x - 3} \over 7} = {{y - 2} \over { - 1}} = {{z - 3} \over { - 4}}$
$(7t + 3) + 2( - t + 2) + 3( - 4t + 3) - 2 = 0$
$ - 7t + 14 = 0$
$t = 2$
So, the required point is $(17,0, - 5)$
$\alpha + \beta + \gamma = 12$
The largest value of $a$, for which the perpendicular distance of the plane containing the lines $ \vec{r}=(\hat{i}+\hat{j})+\lambda(\hat{i}+a \hat{j}-\hat{k})$ and $\vec{r}=(\hat{i}+\hat{j})+\mu(-\hat{i}+\hat{j}-a \hat{k})$ from the point $(2,1,4)$ is $\sqrt{3}$, is _________.
Explanation:
Normal to plane $ = \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 1 & a & { - 1} \cr { - 1} & 1 & { - a} \cr } } \right|$
$ = \widehat i(1 - {a^2}) - \widehat j( - a - 1) + \widehat k(1 + a)$
$ = (1 - a)\widehat i + \widehat j + \widehat k$
$\therefore$ Plane $(1 - a)(x - 1) + (y - 1) + z = 0$
Distance from (2, 1, 4) is $\sqrt 3 $ i.e.
$ \Rightarrow \left| {{{(1 - a) + 0 + 4} \over {\sqrt {{{(1 - a)}^2} + 1 + 1} }}} \right| = \sqrt 3 $
$ \Rightarrow 25 + {a^2} - 10a = 3{a^2} - 6a + 9$
$ \Rightarrow 2{a^2} + 4a - 16 = 0$
$ \Rightarrow {a^2} + 2a - 8 = 0$
$a = 2$ or $ - 4$
$\therefore$ ${a_{\max }} = 2$
The plane passing through the line $L: l x-y+3(1-l) z=1, x+2 y-z=2$ and perpendicular to the plane $3 x+2 y+z=6$ is $3 x-8 y+7 z=4$. If $\theta$ is the acute angle between the line $L$ and the $y$-axis, then $415 \cos ^{2} \theta$ is equal to _____________.
Explanation:
$L:lx - y + 3(1 - l)z = 1$, $x + 2y - z = 2$ and plane containing the line $p:3x - 8y + 7z = 4$
Let $\overrightarrow n $ be the vector parallel to L.
then $\overrightarrow n = \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr l & { - 1} & {3(1 - l)} \cr 1 & 2 & { - 1} \cr } } \right|$
$ = (6l - 5)\widehat i + (3 - 2l)\widehat j + (2l + 1)\widehat k$
$\because$ R containing L
$3(6l - 5) - 8(3 - 2l) + 7(2l + 1) = 0$
$18l + 16l + 14l - 15 - 24 + 7 = 0$
$\therefore$ $l = {{32} \over {48}} = {2 \over 3}$
Let $\theta$ be the acute angle between L and y-axis
$\therefore$ $\cos \theta = {{{5 \over 3}} \over {\sqrt {1 + {{25} \over 9} + {{49} \over 9}} }} = {5 \over {\sqrt {83} }}$
$\therefore$ $415{\cos ^2}\theta = 125$