Question 1

Three vertices of a rectangle AOBC are given as A(0, 2), O(0, 0), and B(4, 0). Calculate the square of the length of its diagonal.

Accepted Answer

Answer: 20. With vertices A(0, 2), O(0, 0), and B(4, 0), the fourth vertex of the rectangle is C(4, 2). The length of the diagonal OC is found using the distance formula: √((4-0)² + (2-0)²) = √(16 + 4) = √20. The square of the diagonal's length is 20.

Question 2

What is the distance of the point (-1, 7) from the x-axis?

Accepted Answer

Answer: 7. The distance of any point (x, y) from the x-axis is the absolute value of its y-coordinate. For the point (-1, 7), this distance is |7| = 7.

Question 3

Consider the following statements:
Assertion (A): The point P(0, 2) is the point of intersection of the y-axis with the line 3x + 2y = 4.
Reason (R): The distance of point P(0, 2) from the x-axis is 2 units.
Which of the following options is correct?

Accepted Answer

Answer: Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).. Assertion (A) is true because substituting x=0 (for y-axis intersection) into 3x + 2y = 4 gives 2y=4, so y=2. Reason (R) is also true as the distance of (0, 2) from the x-axis is |2|=2. However, the reason about distance does not explain why the point lies on the given line, making them independent true statements.

Question 4

Determine the distance between the point (-6, 8) and the origin.

Accepted Answer

Answer: 10. The distance from the origin (0,0) to a point (x,y) is calculated using the distance formula as sqrt(x^2 + y^2). For the point (-6, 8), this is sqrt((-6)^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10.

Question 5

Identify the type of triangle formed by the vertices (-4, 0), (4, 0), and (0, 3).

Accepted Answer

Answer: isosceles triangle. By calculating the lengths of the sides using the distance formula, we find two sides are equal (5 units each), while the third side is different (8 units). A triangle with two equal sides is an isosceles triangle.

Question 6

Find the coordinates of the point on the x-axis that is equidistant from point P(5, 0) and point Q(-1, 0).

Accepted Answer

Answer: (2, 0). Let the point on the x-axis be R(x, 0). The distance PR must equal QR. Using the distance formula, (x-5)^2 = (x+1)^2, which simplifies to x^2 - 10x + 25 = x^2 + 2x + 1. Solving for x gives -12x = -24, so x = 2. The point is (2, 0).

Question 7

A point P has an x-coordinate that is double its y-coordinate. If this point P is equally distant from point Q(2, -5) and point R(-3, 6), what are the coordinates of P?

Accepted Answer

Answer: (16, 8). Let the coordinates of P be (2y, y). Since P is equidistant from Q(2, -5) and R(-3, 6), PQ^2 = PR^2. Using the distance formula, (2y-2)^2 + (y+5)^2 = (2y+3)^2 + (y-6)^2. Expanding and simplifying this equation leads to y=8. Therefore, the coordinates of P are (2*8, 8), which is (16, 8).

Question 8

Calculate the distance between the origin (0, 0) and the point with coordinates (a - b, a + b).

Accepted Answer

Answer: √(2a² + 2b²). The distance from the origin to (x,y) is √(x²+y²). Here, it is √((a-b)² + (a+b)²) = √(a²-2ab+b² + a²+2ab+b²) = √(2a² + 2b²).

Question 9

What is the distance between the points (m, -n) and (-m, n)?

Accepted Answer

Answer: 2√(m² + n²). Applying the distance formula, d = √((-m-m)² + (n-(-n))²) = √((-2m)² + (2n)²) = √(4m² + 4n²) = 2√(m² + n²).

Question 10

Find the coordinates of the point on the x-axis that is equidistant from the points (-4, 0) and (10, 0).

Accepted Answer

Answer: (3, 0). A point on the x-axis equidistant from two points with the same y-coordinate is their midpoint. The midpoint of (-4, 0) and (10, 0) is ((-4+10)/2, 0) = (3, 0).

Class 10 Maths: Distance Formula — Practice Questions with Answers

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