A variable plane passes through a fixed point (a, b, c) and cuts the axes in A, B and C respectively. The locus of the centre of the sphere OABC, O being the origin, is

• $\frac{\mathrm{x}}{\mathrm{a}} + \frac{\mathrm{y}}{\mathrm{b}} + \frac{\mathrm{z}}{\mathrm{c}} = 1$

• $\frac{\mathrm{a}}{\mathrm{x}} + \frac{\mathrm{b}}{\mathrm{y}} + \frac{\mathrm{c}}{\mathrm{z}} = 1$

• $\frac{\mathrm{a}}{\mathrm{x}} + \frac{\mathrm{b}}{\mathrm{y}} + \frac{\mathrm{c}}{\mathrm{z}} = 2$

• $\frac{\mathrm{x}}{\mathrm{a}} + \frac{\mathrm{y}}{\mathrm{b}} +\hspace{0.17em}\frac{\mathrm{z}}{\mathrm{c}} = 2$

The equation of the plane passing through the line of intersection of the planes x + y + z = 1, 2x + 3y + 4z = 7 and perpendicular to the piane x - 5y + 3z = 5 is given by :

• x + 2y + 3z - 6 = 0

• x + 2y + 3z + 6 = 0

• x + 4y + 5z - 8 = 0

• 3x + 4y + 5z + 8 = 0

A plane P passes through the line of intersection of the planes 2x - y + 3z = 2, x + y - z = 1 and the point(1, 0, 1). What are the direction ratios of the line of intersection of the.given planes ?

• (2, - 5, - 3)

• (1, - 5, - 3)

• (2, 5, 3)

• (1, 3, 5)

A plane P passes through the line of intersection of the planes 2x - y + 3z = 2, x + y - z = 1 and the point(1, 0, 1). What is the equation of the plane P ?

• 2x + 5y - 2 = 0

• 5x + 2y - 5 = 0

• x + y - 2 = 0

• 2x - y - 2z = 0

A plane P passes through the line of intersection of the planes 2x - y + 3z = 2, x + y - z = 1 and the point(1, 0, 1). If the plane P touches the sphere x² + y + z² = r, then what is r equal to ?

• $\frac{2}{\sqrt{29}}$

• $\frac{4}{\sqrt{29}}$

• $\frac{5}{\sqrt{29}}$

• 1

Let Q be the image of the point P ( - 2, 1, - 5) in the plane 3x - 2y + 2z + 1 = 0.

Consider the following :

1. The coordinates of Q are (4, - 3, - 1).
2. PQ is of length more than 8 units.
3. The point (1, - 1, - 3) is the mid-point of the line segment PQ and lies on the given plane. Which of the above statements are correct ?

• 1 and 2 only

• 2 and 3 only

• 1 and 3 only

• 1, 2 and 3

Let Q be the image of the point P ( - 2, 1, - 5) in the plane 3x - 2y + 2z + 1 = 0.

Consider the following :

1. The direction ratios of the line segment PQ are < 3, - 2, 2 >.
2. The sum of the squares of direction cosines of the line segment PQ is unity.

Which of the above statements is/are correct ?

• 1 only

• 2 only

• Both 1 and 2

• Neither 1 nor 2

A line L passes through the point P(5, - 6, 7) and is parallel to the planes x + y + z = 1 and 2x - y - 2z = 3.

What are the direction ratios of the line of intersection of the given planes ?

• < 1, 4, 3 >

• < - 1, - 4, 3 >

• < 1, - 4, 3 >

• < 1, - 4, - 3 >

19.

A line L passes through the point P(5, - 6, 7) and is parallel to the planes x + y + z = 1 and 2x - y - 2z = 3.

What is the equation of the line ?

• $\frac{\mathrm{x} - 5}{ - 1} = \frac{\mathrm{y} + 6}{4} = \frac{\mathrm{z} - 7}{ - 3}$

• $\frac{\mathrm{x} + 5}{ - 1} = \frac{\mathrm{y} - 6}{4} - \frac{\mathrm{z} + 7}{ - 3}$

• $\frac{\mathrm{x} - 5}{ - 1} = \frac{\mathrm{y} + 6}{4} - \frac{\mathrm{z} - 7}{ - 3}$

• $\frac{\mathrm{x} - 5}{ - 1} = \frac{\mathrm{y} + 6}{ - 4} - \frac{\mathrm{z} - 7}{ - 3}$

A straight line passes through the point (1, 1, 1) makes an angle 60^o with the positive direction of z-axis, and the cosine of the angles made by it with the positive directions of the y-axis and the x-axis are in the ration $\sqrt{3} : 1$. What is the acute angle between the two possible positions of the line ?

• $90°$

• $60°$

• $45°$

• $30°$