SENAPREP
3D points Questions
Chapter: Analytical Geometry of Two & Three-Dimensions | Total Available: 14 Real Questions
Consider the points A(2, 4, 6), B(-2, -4, -2), C(4, 6, 4) andD(8, 14, 12).
Which of the following statements is/are correct? 1. The points are the vertices of a rectangle ABCD.
2. The mid-point of AC is the same as that of BD.
Select the correct answer using the code given below:
The number of spheres of radius r touching the coordinate axes is
The xy-plane divides the line segment joining the points (-1, 3, 4) and (2, -5, 6)
The locus of a point P(x, y, z) which moves in such a way that z = 7 is a
Explanation : Given, point z = 7
What is the number of possible values of k for which the line joining the points (k, 1, 3) and (1, - 2, k + 1) also passes through the point (15, 2, - 4)??
What is the projection of the line segment joining (1, 7, -5) and B (-3, 4, -2) on y-axis?
For the next two (02) items that follow:
P(−2, −3, 5), Q(4, −1, 5), R(6, −4, 8) and S(2, −6, 10) are four points.
What is (PQ² + 2QS² − 2PR²) equal to?
Solution
Option (b) is correct.
Explanation:
To solve for \(PQ^2 + 2QS^2 - 2PR^2\), we compute the distances first. The distance formula in 3D: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \] Calculating \(PQ\): \[ PQ = \sqrt{(4 - (-2))^2 + (-1 - (-3))^2 + (5 - 5)^2} \] \[ = \sqrt{(6)^2 + (2)^2 + 0} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10} \] \[ PQ^2 = (2\sqrt{10})^2 = 40 \] Calculating \(QS\): \[ QS = \sqrt{(2 - 4)^2 + (-6 - (-1))^2 + (10 - 5)^2} \] \[ = \sqrt{(-2)^2 + (-5)^2 + (5)^2} = \sqrt{4 + 25 + 25} = \sqrt{54} = 3\sqrt{6} \] \[ QS^2 = (3\sqrt{6})^2 = 54 \] Calculating \(PR\): \[ PR = \sqrt{(6 - (-2))^2 + (-4 - (-3))^2 + (8 - 5)^2} \] \[ = \sqrt{(8)^2 + (-1)^2 + (3)^2} = \sqrt{64 + 1 + 9} = \sqrt{74} \] \[ PR^2 = 74 \] Now substituting into the expression: \[ PQ^2 + 2QS^2 - 2PR^2 = 40 + 2 \cdot 54 - 2 \cdot 74 \] \[ = 40 + 108 - 148 = 0 \] Thus, the final answer is \(0\).- 3D Distance Formula.
- Properties of squares in distances.
A 3D coordinate system diagram showing points P, Q, R, and S, with lines connecting them, would clarify position relationships and distances.
Verified against NCERT Class XI Mathematics, Chapter 11 (Conic Sections).
For the next two (02) items that follow:
P(−2, −3, 5), Q(4, −1, 5), R(6, −4, 8) and S(2, −6, 10) are four points.
Consider the following statements:
I. PQ is parallel to RS.
II. PR is perpendicular to QS.
Which of the statements given above is/are correct?
Solution
Option (d) is correct.
Explanation:
To determine the correctness of statements I and II, we first check the slopes of the line segments. 1. For line segment PQ: - Coordinates: P(-2, -3) and Q(4, -1) - Slope (m_PQ):- Slope of a line: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
- Two lines are parallel if their slopes are equal.
- Two lines are perpendicular if the product of their slopes is -1.
Utilize a coordinate plane to plot points P, Q, R, and S. Draw line segments PQ and RS, indicating their slopes, and label angles to show potential perpendicularity.
Verified against NCERT Class XI Mathematics, Coordinate Geometry.
For the following two (02) items:
Let A(1, –1, 0), B(-2, 1, 8) and C(-1, 2, 7) are three consecutive vertices of a parallelogram ABCD.
If angle BCD is θ, then what is cos2θequal to?
For the following two (02) items:
Let A(1, –1, 0), B(-2, 1, 8) and C(-1, 2, 7) are three consecutive vertices of a parallelogram ABCD.
What is the fourth vertex D?
