SCERT Maharashtra solutions for Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC chapter 1.6 - Line and Plane [Latest edition]

The equation of X axis is ______ 

The perpendicular distance of the plane 2x + 3y – z = k from the origin is ${\displaystyle \sqrt{14}}$ units, the value of k is ______.

Choose correct alternatives :

The equation of the plane passing through the points (1, −1, 1), (3, 2, 4) and parallel to the Y-axis is ______  

The direction ratios of the line 3x + 1 = 6y – 2 = 1 – z are ______.

If the planes 2x – my + z = 3 and 4x – y + 2z = 5 are parallel then m = ______ 

Choose correct alternatives :

The direction cosines of the normal to the plane 2x – y + 2z = 3 are ______ 

If the foot of the perpendicular drawn from the origin to the plane is (4, −2, -5), then the equation of the plane is ______ 

The perpendicular distance of the origin from the plane x − 3y + 4z = 6 is ______ 

The coordinates of the foot of perpendicular drawn from the origin to the plane 2x + y − 2z = 18 are ______ 

Find the cartesian equation of the plane passing through A(1, 2, 3) and the direction ratios of whose normal are 3, 2, 5.

Find the direction ratios of the normal to the plane 2x + 3y + z = 7

Find the vector equation of the line ${\displaystyle \frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}}$

Verify if the point having position vector ${\displaystyle 4\hat{\text{i}}-11\hat{\text{j}}+2\hat{\text{k}}}$ lies on the line ${\displaystyle \overline{\text{r}}=(6\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}})+\lambda (2\hat{\text{i}}+7\hat{\text{j}}+3\hat{\text{k}})}$

Find the Cartesian equation of the line passing through  A(1, 2, 3) and having direction ratios 2, 3, 7

Find the vector equation of the line passing through the point having position vector ${\displaystyle 4\hat{i}-\hat{j}+2\hat{\text{k}}}$ and parallel to the vector ${\displaystyle -2\hat{i}-\hat{j}+\hat{k}}$.

Find the Cartesian equation of the plane passing through the points (3, 2, 1) and (1, 3, 1)

Find the direction ratios of the line perpendicular to the lines

${\displaystyle \frac{x-7}{2}=\frac{y+7}{-3}=\frac{z-6}{1}}$ and ${\displaystyle \frac{x+5}{1}=\frac{y+3}{2}=\frac{z-6}{-2}}$

Find direction cosines of the normal to the plane ${\displaystyle \overline{\text{r}}\cdot (3\hat{\text{i}}+4\hat{\text{k}})}$ = 5

If the normal to the plane has direction ratios 2, −1, 2 and it’s perpendicular distance from origin is 6, find its equation

Reduce the equation ${\displaystyle \overline{\text{r}}\cdot (3\hat{\text{i}}+4\hat{\text{j}}+12\hat{\text{k}})}$ = 8 to normal form

Find the Cartesian equation of the line passing through A(1, 2, 3) and B(2, 3, 4)

Find the perpendicular distance of origin from the plane 6x − 2y + 3z - 7 = 0

Find the acute angle between the lines x = y, z = 0 and x = 0, z = 0

Find Cartesian equation of the line passing through the point A(2, 1, −3) and perpendicular to vectors ${\displaystyle \hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}}$ and ${\displaystyle \hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}}}$

Find the vector equation of the line passing through the point having position vector ${\displaystyle -\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}}}$ and parallel to the line ${\displaystyle \overline{\text{r}}=(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})+\mu (3\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}})}$, µ is a parameter

Find the Cartesian equation of the line passing through (−1, −1, 2) and parallel to the line 2x − 2 = 3y + 1 = 6z – 2

Find the Cartesian equation of the plane passing through A(7, 8, 6)and parallel to XY plane

Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x + 6y – 3z = 63.

Find the vector equation of a plane at a distance 6 units from the origin and to which vector ${\displaystyle 2\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}}}$ is normal

Find the Cartesian equation of the plane passing through the points A(1, 1, 2), B(0, 2, 3) C(4, 5, 6)

Find acute angle between the lines ${\displaystyle \frac{x-1}{1}=\frac{y-2}{-1}=\frac{z-3}{2}}$ and ${\displaystyle \frac{x-1}{2}=\frac{y-1}{1}=\frac{z-3}{1}}$

Find the distance between the parallel lines ${\displaystyle \frac{x}{2}=\frac{y}{-1}=\frac{z}{2}}$ and ${\displaystyle \frac{x-1}{2}=\frac{y-1}{-1}=\frac{z-1}{2}}$

Find the equation of the plane passing through the point (7, 8, 6) and parallel to the plane ${\displaystyle \overline{\text{r}}\cdot (6\hat{\text{i}}+8\hat{\text{j}}+7\hat{\text{k}})}$ = 0

Find m, if the lines ${\displaystyle \frac{1-x}{3}=\frac{7y-14}{2\text{m}}=\frac{z-3}{2}}$ and ${\displaystyle \frac{7-7x}{3\text{m}}=\frac{y-5}{1}=\frac{6-z}{5}}$ are at right angles

Show that the lines ${\displaystyle \frac{x+1}{-10}=\frac{y+3}{-1}=\frac{z-4}{1}}$ and ${\displaystyle \frac{x+10}{-1}=\frac{y+1}{-3}=\frac{z-1}{4}}$ intersect each other.also find the coordinates of the point of intersection

A(– 2, 3, 4), B(1, 1, 2) and C(4, –1, 0) are three points. Find the Cartesian equations of the line AB and show that points A, B, C are collinear.

Find the Cartesian and vector equation of the line passing through the point having position vector ${\displaystyle \hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}}$ and perpendicular to vectors ${\displaystyle \hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}}$ and ${\displaystyle 2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}}$

Find the vector equation of the plane which bisects the segment joining A(2, 3, 6) and B(4, 3, −2) at right angles

Find vector equation of the plane passing through A(−2 ,7 ,5) and parallel to vectors ${\displaystyle 4\hat{\text{i}} -\hat{\text{j}}+3\hat{\text{k}}}$ and ${\displaystyle \hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}}$

Find the Cartesian and vector equation of the plane which makes intercepts 1, 1, 1 on the coordinate axes