Question 1

Q1: A point is on the $x$-axis. What are its $y$-coordinate and $z$-coordinates?

Accepted Answer

Concept: Any point lying on the $x$-axis has its perpendicular distance from the YZ-plane as some value $x$, but its perpendicular distances from the XZ-plane and XY-plane are zero. The distance from the XZ-plane is the $y$-coordinate, and the distance from the XY-plane is the $z$-coordinate. Soluti...

Question 2

Q2: A point is in the XZ-plane. What can you say about its $y$-coordinate?

Accepted Answer

Concept: Any point lying in the XZ-plane has its perpendicular distance from the XZ-plane equal to zero. The perpendicular distance of a point $(x, y, z)$ from the XZ-plane is given by its $y$-coordinate. Solution: For any point in the XZ-plane, its coordinates are of the form $(x, 0, z)$. Therefore...

Question 3

Q3: Name the octants in which the following points lie: $(1,2,3),(4,-2,3),(4,-2,-5),(4,2,-5),(-4,2,-5),(-4,2,5)$, $(-3,-1,6)(-2,-4,-7)$.

Accepted Answer

Concept: The octant in which a point lies is determined by the signs of its $x, y,$ and $z$ coordinates. - Octant I: $(+,+,+)$ - Octant II: $(-,+,+)$ - Octant III: $(-,-,+)$ - Octant IV: $(+,-,+)$ - Octant V: $(+,+,-)$ - Octant VI: $(-,+,-)$ - Octant VII: $(-,-,-)$ - Octant VIII: $(+,-,-)$ Solution:...

Question 4

Q4: Fill in the blanks: (i) The $x$-axis and $y$-axis taken together determine a plane known as _____ . (ii) The coordinates of points in the XY-plane are...

Accepted Answer

Solution: (i) The $x$-axis and $y$-axis taken together determine a plane known as the XY-plane. (ii) The coordinates of points in the XY-plane are of the form (x, y, 0), because the $z$-coordinate, which represents the distance from the XY-plane, is zero for any point on it. (iii) The three mutually...

Question 5

Q1: Find the distance between the following pairs of points: (i) $(2,3,5)$ and $(4,3,1)$ (ii) $(-3,7,2)$ and $(2,4,-1)$ (iii) $(-1,3,-4)$ and $(1,-3,4)$ (...

Accepted Answer

Formula: The distance between two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ is given by: $$PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$ (i) Points: $(2,3,5)$ and $(4,3,1)$ Solution: $$d = \sqrt{(4-2)^2 + (3-3)^2 + (1-5)^2}$$ $$d = \sqrt{2^2 + 0^2 + (-4)^2}$$ $$d = \sqrt{4 + 0 +...

Question 6

Q2: Show that the points $(-2,3,5),(1,2,3)$ and $(7,0,-1)$ are collinear.

Accepted Answer

To Show: The points P(-2,3,5), Q(1,2,3), and R(7,0,-1) are collinear. Concept: Three points are collinear if they lie on the same straight line. This can be verified by showing that the sum of the distances between two pairs of points is equal to the distance between the third pair. That is, $PQ + Q...

Question 7

Q3: Verify the following: (i) $(0,7,-10),(1,6,-6)$ and $(4,9,-6)$ are the vertices of an isosceles triangle. (ii) $(0,7,10),(-1,6,6)$ and $(-4,9,6)$ are t...

Accepted Answer

(i) Verify that $(0,7,-10),(1,6,-6)$ and $(4,9,-6)$ are the vertices of an isosceles triangle. Given: Vertices A(0, 7, -10), B(1, 6, -6), and C(4, 9, -6). To Verify: Triangle ABC is an isosceles triangle. Solution: We calculate the lengths of the three sides of the triangle. $$AB = \sqrt{(1-0)^2 + (...

Question 8

Q4: Find the equation of the set of points which are equidistant from the points $(1,2,3)$ and $(3,2,-1)$.

Accepted Answer

Given: Let the points be A(1, 2, 3) and B(3, 2, -1). Let P(x, y, z) be any point on the required set. Condition: The point P is equidistant from A and B, which means PA = PB. This implies $PA^2 = PB^2$. Formula: The square of the distance between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is...

Question 9

Q5: Find the equation of the set of points P, the sum of whose distances from A(4,0,0) and B(-4,0,0) is equal to 10.

Accepted Answer

Given: Points A(4, 0, 0) and B(-4, 0, 0). Let P(x, y, z) be any point in the set. Condition: The sum of the distances from P to A and B is 10. That is, PA + PB = 10. Solution: Using the distance formula: $$PA = \sqrt{(x-4)^2 + (y-0)^2 + (z-0)^2} = \sqrt{(x-4)^2 + y^2 + z^2}$$ $$PB = \sqrt{(x-(-4))^2...

Question 10

Q1: Three vertices of a parallelogram ABCD are A(3,-1,2), B(1,2,-4) and C(-1,1,2). Find the coordinates of the fourth vertex.

Accepted Answer

Given: Vertices of a parallelogram ABCD are A(3, -1, 2), B(1, 2, -4), and C(-1, 1, 2). To Find: The coordinates of the fourth vertex D(x, y, z). Concept: In a parallelogram, the diagonals bisect each other. This means the midpoint of the diagonal AC is the same as the midpoint of the diagonal BD. Fo...

Question 11

Q2: Find the lengths of the medians of the triangle with vertices A(0,0,6), B(0,4,0) and C(6,0,0).

Accepted Answer

Given: The vertices of a triangle are A(0, 0, 6), B(0, 4, 0), and C(6, 0, 0). To Find: The lengths of the three medians of the triangle. Concept: A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Solution: First, we find the midpoints of the sides BC, AC...

Question 12

Q3: If the origin is the centroid of the triangle PQR with vertices P(2a, 2, 6), Q(-4, 3b, -10) and R(8, 14, 2c), then find the values of a, b and c.

Accepted Answer

Given: The vertices of triangle PQR are P(2a, 2, 6), Q(-4, 3b, -10), and R(8, 14, 2c). The centroid of the triangle is the origin, G(0, 0, 0). To Find: The values of a, b, and c. Formula: The coordinates of the centroid G of a triangle with vertices $(x_1, y_1, z_1), (x_2, y_2, z_2), (x_3, y_3, z_3)...

Question 13

Q4: If A and B be the points (3,4,5) and (-1,3,-7), respectively, find the equation of the set of points P such that $PA^2 + PB^2 = k^2$, where $k$ is a c...

Accepted Answer

Given: Points A(3, 4, 5) and B(-1, 3, -7). Let P(x, y, z) be any point in the set. Condition: $PA^2 + PB^2 = k^2$, where $k$ is a constant. Formula: The square of the distance between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is $(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2$. Solution: Fir...

NCERT Solutions