3D Geometry

Let \(\alpha x+\beta y+\gamma z=1\) be the equation of a plane passing through the point \((3,-2,5)\) and perpendicular to the line joining the points \((1,2,3)\) and \((-2,3,5)\). Then the value of \(\alpha \beta y\) is equal to __________...

The point of intersection \(\mathrm{C}\) of the plane \(8 x+y+2 z=0\) and the line joining the points \(\mathrm{A}(-3,-6,1)\) and \(\mathrm{B}(2,4,-3)\) divides the line segment \(\mathrm{AB}\) internally in the ratio \(\mathrm{k}: 1\). If ...

Let the plane P pass through the intersection of the planes \(2x+3y-z=2\) and \(x+2y+3z=6\), and be perpendicular to the plane \(2x+y-z+1=0\). If d is the distance of P from the point (\(-\)7, 1, 1), then \(\mathrm{d^{2}}\) is equal to :

Let the plane $P$ contain the line $2 x+y-z-3=0=5 x-3 y+4 z+9$ and be parallel to the line $\frac{x+2}{2}=\frac{3-y}{-4}=\frac{z-7}{5}$. Then the distance of the point $\mathrm{A}(8,-1,-19)$ from the plane $\mathrm{P}$ measured parallel to ...

Let $\mathrm{S}$ be the set of all values of $\lambda$, for which the shortest distance between the lines $\frac{x-\lambda}{0}=\frac{y-3}{4}=\frac{z+6}{1}$ and $\frac{x+\lambda}{3}=\frac{y}{-4}=\frac{z-6}{0}$ is 13. Then $8\left|\sum\limits...

Let the system of linear equations
$-x+2 y-9 z=7$
$-x+3 y+7 z=9$
$-2 x+y+5 z=8$
$-3 x+y+13 z=\lambda$

has a unique solution $x=\alpha, y=\beta, z=\gamma$. Then the distance of the point

$(\alpha, \beta, \gamma)$ from the plane $2 x-2 y+z=...

Let the foot of perpendicular of the point $P(3,-2,-9)$ on the plane passing through the points $(-1,-2,-3),(9,3,4),(9,-2,1)$ be $Q(\alpha, \beta, \gamma)$. Then the distance of $Q$ from the origin is :

Let the image of the point \(\left(\frac{5}{3}, \frac{5}{3}, \frac{8}{3}\right)\) in the plane \(x-2 y+z-2=0\) be P. If the distance of the point \(Q(6,-2, \alpha), \alpha > 0\), from \(\mathrm{P}\) is 13 , then \(\alpha\) is equal to _____...

The distance of the point \((-1,2,3)\) from the plane \(\vec{r} \cdot(\hat{i}-2 \hat{j}+3 \hat{k})=10\) parallel to the line of the shortest distance between the lines \(\vec{r}=(\hat{i}-\hat{j})+\lambda(2 \hat{i}+\hat{k})\) and $$\vec{r}=(...

Let the equation of plane passing through the line of intersection of the planes \(x+2 y+a z=2\) and \(x-y+z=3\) be \(5 x-11 y+b z=6 a-1\). For \(c \in \mathbb{Z}\), if the distance of this plane from the point \((a,-c, c)\) is $$\frac{2}{\...

Let \(\mathrm{N}\) be the foot of perpendicular from the point \(\mathrm{P}(1,-2,3)\) on the line passing through the points \((4,5,8)\) and \((1,-7,5)\). Then the distance of \(N\) from the plane \(2 x-2 y+z+5=0\) is :

The plane, passing through the points \((0,-1,2)\) and \((-1,2,1)\) and parallel to the line passing through \((5,1,-7)\) and \((1,-1,-1)\), also passes through the point :

The line, that is coplanar to the line \(\frac{x+3}{-3}=\frac{y-1}{1}=\frac{z-5}{5}\), is :

Let the plane \(x+3 y-2 z+6=0\) meet the co-ordinate axes at the points A, B, C. If the orthocenter of the triangle \(\mathrm{ABC}\) is \(\left(\alpha, \beta, \frac{6}{7}\right)\), then \(98(\alpha+\beta)^{2}\) is equal to ___________.

Let the plane P: \(4 x-y+z=10\) be rotated by an angle \(\frac{\pi}{2}\) about its line of intersection with the plane \(x+y-z=4\). If \(\alpha\) is the distance of the point \((2,3,-4)\) from the new position of the plane \(\mathrm{P}\), t...

Let the lines \(l_{1}: \frac{x+5}{3}=\frac{y+4}{1}=\frac{z-\alpha}{-2}\) and \(l_{2}: 3 x+2 y+z-2=0=x-3 y+2 z-13\) be coplanar. If the point \(\mathrm{P}(a, b, c)\) on \(l_{1}\) is nearest to the point \(\mathrm{Q}(-4,-3,2)\), then $$|a|+|b...

Let a line \(l\) pass through the origin and be perpendicular to the lines
\(l_{1}: \vec{r}=(\hat{\imath}-11 \hat{\jmath}-7 \hat{k})+\lambda(\hat{i}+2 \hat{\jmath}+3 \hat{k}), \lambda \in \mathbb{R}\) and
$$l_{2}: \vec{r}=(-\hat{\imath}+\ha...

If equation of the plane that contains the point \((-2,3,5)\) and is perpendicular to each of the planes \(2 x+4 y+5 z=8\) and \(3 x-2 y+3 z=5\) is \(\alpha x+\beta y+\gamma z+97=0\) then \(\alpha+\beta+\gamma=\)

Let \((\alpha, \beta, \gamma)\) be the image of the point \(\mathrm{P}(2,3,5)\) in the plane \(2 x+y-3 z=6\). Then \(\alpha+\beta+\gamma\) is equal to :

Let the line \(l: x=\frac{1-y}{-2}=\frac{z-3}{\lambda}, \lambda \in \mathbb{R}\) meet the plane \(P: x+2 y+3 z=4\) at the point \((\alpha, \beta, \gamma)\). If the angle between the line \(l\) and the plane \(P\) is $$\cos ^{-1}\left(\sqrt{...

Let P be the plane passing through the points \((5,3,0),(13,3,-2)\) and \((1,6,2)\).

For \(\alpha \in \mathbb{N}\), if the distances of the points \(\mathrm{A}(3,4, \alpha)\) and \(\mathrm{B}(2, \alpha, a)\) from the plane P are 2 and 3 re...

Let the line passing through the points \(\mathrm{P}(2,-1,2)\) and \(\mathrm{Q}(5,3,4)\) meet the plane \(x-y+z=4\) at the point \(\mathrm{R}\). Then the distance of the point \(\mathrm{R}\) from the plane \(x+2 y+3 z+2=0\) measured paralle...

Let P be the point of intersection of the line \({{x + 3} \over 3} = {{y + 2} \over 1} = {{1 - z} \over 2}\) and the plane \(x+y+z=2\). If the distance of the point P from the plane \(3x - 4y + 12z = 32\) is q, then q and 2q are the roots o...

Let two vertices of a triangle ABC be (2, 4, 6) and (0, \(-\)2, \(-\)5), and its centroid be (2, 1, \(-\)1). If the image of the third vertex in the plane \(x+2y+4z=11\) is \((\alpha,\beta,\gamma)\), then $$\alpha\beta+\beta\gamma+\gamma\al...

The shortest distance between the lines \({{x + 2} \over 1} = {y \over { - 2}} = {{z - 5} \over 2}\) and \({{x - 4} \over 1} = {{y - 1} \over 2} = {{z + 3} \over 0}\) is :

Let the foot of perpendicular from the point \(\mathrm{A}(4,3,1)\) on the plane \(\mathrm{P}: x-y+2 z+3=0\) be N. If B\((5, \alpha, \beta), \alpha, \beta \in \mathbb{Z}\) is a point on plane P such that the area of the triangle ABN is $$3 \...

Let the line \(\frac{x}{1}=\frac{6-y}{2}=\frac{z+8}{5}\) intersect the lines \(\frac{x-5}{4}=\frac{y-7}{3}=\frac{z+2}{1}\) and \(\frac{x+3}{6}=\frac{3-y}{3}=\frac{z-6}{1}\) at the points \(\mathrm{A}\) and \(\mathrm{B}\) respectively. Then ...

Let the image of the point \(\mathrm{P}(1,2,6)\) in the plane passing through the points \(\mathrm{A}(1,2,0), \mathrm{B}(1,4,1)\) and \(\mathrm{C}(0,5,1)\) be \(\mathrm{Q}(\alpha, \beta, \gamma)\). Then $$\left(\alpha^{2}+\beta^{2}+\gamma^{...

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 $$\De...

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 \({P_1}:\overrightarrow r \,.\,\left( {2\widehat i + \widehat j - 3\widehat k} \right) = 4\) be a plane. Let P2 be another plane which passes through the points (2, \(-\)3, 2), (2, \(-\)2, \(-\)3) and (1, \(-\)4, 2). If the direction ra...

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 d2 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 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 ...

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 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 th...

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, measur...

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:

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 :

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...

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 :

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 :

Let the image of the point P(1, 2, 3) in the line \(L:{{x - 6} \over 3} = {{y - 1} \over 2} = {{z - 2} \over 3}\) be Q. Let R (\(\alpha\), \(\beta\), \(\gamma\)) be a point that divides internally the line segment PQ in the ratio 1 : 3. The...

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 + 20...

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 $$\b...

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?

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 intercept...

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 NO...

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 ______________.

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 :