RS Aggarwal

### Find the angle between each of the following pairs of lines:

To find – Angle between the two pair of lines Direction ratios of L1 = (-3,-2,0) Direction ratios of L2 = (1,-3,2) Tip – If (a,b,c) be the direction ratios of the first line and (a’,b’,c’) be that...

### Find the coordinates of the foot of the perpendicular drawn from the point (1, 2, 3) to the line

Find the coordinates of the foot of the perpendicular drawn from the point $(1,2,3)$ to the line $\frac{\mathrm{x}-6}{3}=\frac{\mathrm{y}-7}{2}=\frac{\mathrm{z}-7}{-2}$. Also, find the length of the...

### Show that the lines and do not intersect each other.

Show that the lines $\frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}+1}{3}=z$ and $\frac{\mathrm{x}+1}{5}=\frac{\mathrm{y}-2}{1}, Z=2$ do not intersect each other. Answer Given: The equations of the two...

### Find the angle between the line and the plane Answer: The equation of line is: r→=(3i^+k^)+λ(j^+k^) \vec{r}=(3 \hat{i}+\hat{k})+\lambda(\hat {j}+\hat{k}) Comparing with $\vec{r}=\vec{a}+\lambda \vec{b}$ $\vec{b}=(j+\dot{k})$. The...

### Find the angle between the line and the plane Answer: The given line is $\frac{x-2}{3}=\frac{y+1}{-1}=\frac{2-3}{2}$ ⇒r→=(2i→-j→+k→)+t(3i→-j→-2k→) \Rightarrow...

### Prove that the lines are coplanar. Also find the equation of the plane containing these lines.

Answer:                       - 7x + 14 + 14y – 56 – 7z + 42 = 0 - 7x + 14y – 7z = 0 Multiply by negative sign, x – 2y + z = 0 The equation of...

### Find the (i) lengths of the axes, (ii) coordinates of the vertices Given Equation: $\frac{{{x}^{2}}}{9}-\frac{{{y}^{2}}}{16}=1$ Comparing with the equation of hyperbola $\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1$ we get, a = 3 and b = 4 (i)...

### Find the value of λ for which the given planes are perpendicular to each other: (i) and (ii) and .

Answer: (i) For plane perpendicular, cos 900 = 0                   (ii) For plane perpendicular, cos 900 = 0         (λ × 1) + (2 ×...

### Show that the following planes are at right angles: (i) and (ii) and .

Answer: (i)                   (ii)                   = 6 + 24 – 30 = 30 – 30 = 0 = RHS Hence, the planes are...

### Find the acute angle between the following planes: (i) 2x – y + z = 5 and x + y + 2z = 7 (ii) x + 2y + 2z = 3 and 2x – 3y + 6z = 8

Answer: (i) The standard equation of plane,                               (ii) The standard equation of plane,  ...

### Find the acute angle between the following planes: (i) x + y – z = 4 and x + 2y + z = 9 (ii) x + y – 2z = 6 and 2x – 2y + z = 11

Answer: (i) The standard equation of plane,                             (ii) The standard equation of plane,    ...

### Show that each of the following pairs of planes are at right angles: (i) 3x + 4y – 5z = 7 and 2x + 6y + 6z + 7 = 0 (ii) x – 2y + 4z = 10 and 18x + 17y + 4z = 49

Answer: (i) if θ = 900 then cos 900 = 0 A1A2 + B1B2 + C1C2 = 0 By comparing with the standard equation of a plane, A1 = 3, B1 = 4, C1 = -5 A2 = 2, B2 = 6, C2 = 6 LHS = A1A2 + B1B2 + C1C2 = (3 × 2) +...

### Find the equation of the plane passing through the intersection of the planes x – 2y + z = 1 and 2x + y + z = 8, and parallel to the line with direction ratios 1, 2, 1. Also, find the perpendicular distance of (1, 1, 1) from the plane.

Answer: Cartesian form of equation of plane through the line of intersection of planes:                          ...

### Find the equation of the plane passing through the line of intersection of the planes 2x – y = 0 and 3z – y = 0, and perpendicular to the plane 4x + 5y – 3z = 0.

Answer: Cartesian form of equation of plane through the line of intersection of planes:                             The equation...

### Find the equation of the plane through the line of intersection of the planes x – 3y + z + 6 = 0 and x + 2y + 3z + 5 = 0, and passing through the origin.

Answer: Cartesian form of equation of plane through the line of intersection of planes: - x – 27y – 13z = 0 Multiplying by negative sign, x + 27y + 13z = 0 The equation of the plane is x + 27y + 13z...

### Find the equation of the planes passing through the intersection of the planes 2x + 3y – z + 1 = 0 and x + y – 2z + 3 = 0, and perpendicular to the plane 3x – y – 2z – 4 = 0.

Answer: Cartesian form of equation of plane through the line of intersection of planes:     The equation of the plane is 7x + 13y + 4z = 9.

### Find the equation of the plane through the line of intersection of the planes x + y + z = 6 and 2x + 3y + 4z + 5 = 0, and passing through the point (1, 1, 1).

Answer: Cartesian form of equation of plane through the line of intersection of planes:               14x + 14y + 14z – 84 + 6x + 9y + 12z + 15 = 0 20x + 23y + 26z...

### 5. Find the equation of the plane passing through the point (1, 4, -2) and parallel to the plane 2x – y + 3z + 7 = 0.

Answer: Any plane parallel to 2x – y + 3z + 7 = 0 is shown as 2x – y + 3z + d = 0. It passes through the point (1, 4, -2) \$\begin{array}{l} 2 \times 1 - 4 + 3( - 2) + d = 0\\ 2 - 4 - 6 + d = 0\\ d =...