Selina solutions for Concise Mathematics [English] Class 9 ICSE chapter 25 - Distance Formula [Latest edition]

The distance between the points (7, −5) and (3, −1) is:

The distance of the point (−4, 3) from the origin is ______.

The distance between the points (−3, 2) and (x, 10) is 10 units. The value of x is ______.

The point (x, y) is equidistant from the points (3, 6) and (−3, 4). The relation between x and y is ______.

The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is ______.

Find the distance between the following pairs of points:

(–3, 6) and (2, –6)

Find the distance between the following pairs of points:

(a, b), (−a, −b)

Find the distance between the following pairs of points:
`(3/5,2) and (-(1)/(5),1(2)/(5))`

Find the distance between the following pair of points:

`(sqrt(3)+1,1)` and `(0, sqrt(3))`

Find the distance between the origin and the point:
(-8, 6) 

Find the distance between the origin and the point:
(-5, -12)

Find the distance between the origin and the point:
(8, −15)

The distance between the points (3, 1) and (0, x) is 5. Find x.

Find the co-ordinates of points on the x-axis which are at a distance of 17 units from the point (11, -8).

Find the coordinates of the points on the y-axis, which are at a distance of 10 units from the point (-8, 4).

A point A is at a distance of `sqrt(10)` unit from the point (4, 3). Find the co-ordinates of point A, if its ordinate is twice its abscissa.

A point P (2, -1) is equidistant from the points (a, 7) and (-3, a). Find a.

What point on the x-axis is equidistant from the points (7, 6) and (-3, 4)?

Find a point on the y-axis which is equidistant from the points (5, 2) and (-4, 3).

A point P lies on the x-axis and another point Q lies on the y-axis.
Write the ordinate of point P.

A point P lies on the x-axis and another point Q lies on the y-axis.
Write the abscissa of point Q.

A point P lies on the x-axis and another point Q lies on the y-axis.
If the abscissa of point P is -12 and the ordinate of point Q is -16; calculate the length of line segment PQ.

Show that the points P (0, 5), Q (5, 10) and R (6, 3) are the vertices of an isosceles triangle.

Prove that the points P (0, -4), Q (6, 2), R (3, 5) and S (-3, -1) are the vertices of a rectangle PQRS.

Prove that the points A (1, -3), B (-3, 0) and C (4, 1) are the vertices of an isosceles right-angled triangle. Find the area of the triangle.

Show that the points A (5, 6), B (1, 5), C (2, 1) and D (6, 2) are the vertices of a square ABCD.

The distance of the point (−8, 6) from the x-axis is ______.

The distance of the point (−4, −3) from the origin is:

The coordinates of point P are ______.

AB (= 10 unit) is diameter of a circle with centre at point P = (x, 0) and point B = (0, y). The relation between x and y is:

Statement (1): For the point P, 𝑥 = −4 and 𝑦 = 3, the distance of P from origin is 3 + 4 = 7.

Statement (2): P = (−4, 3) and its distance from origin \[= \sqrt{(-4)^2 + (3)^2}\].

Statement (1): The point P⁡(𝑥, 𝑦) is at a distance of 6 unit from origin, then P lies in the first quadrant.

Statement (2): Point P can lie in any quadrant.

Assertion (A): If A = (2⁢𝑥, 𝑦), B⁡(𝑥, 2⁢𝑦) and AB = 5 unit, then 𝑥 + 𝑦 = 5.

Reason (R): \[\sqrt{(x - 2x)^2 + (2y - y)^2} = 5\] \[\Rightarrow x^2 + y^2 = 25\]

Assertion (A): The distance between the points A⁡(𝑥, 2⁢𝑥) and B⁡(𝑥, 0) is 4 unit, the point B is (2, 0).

Reason (R): \[\sqrt{(x - x)^2 + (2x - 0)^2} = 4\] \[\Rightarrow x^2 = 4 \text{ and } x = \pm 2\]

Find the points on the 𝑦-axis which are at a distance of \[2\sqrt{5}\] units from the point (−4, 7).

Find the value of k, if the point P (2, 4) is equidistant from the points A(5, k) and B (k, 7).

The centre of a circle is (2a, a – 7). Find the values of a if the circle passes through the point (11, –9) and has diameter `10sqrt(2)` units.

Show that (-3, 2), (-5, -5), (2, -3) and (4, 4) are the vertices of a rhombus.

Points A (-3, -2), B (-6, a), C (-3, -4) and D (0, -1) are the vertices of quadrilateral ABCD; find a if 'a' is negative and AB = CD.

The vertices of a triangle are (5, 1), (11, 1) and (11, 9). Find the co-ordinates of the circumcentre of the triangle.

Given A = (3, 1) and B = (0, y - 1). Find y if AB = 5.

Given A = (x + 2, -2) and B (11, 6). Find x if AB = 17.

The centre of a circle is (2x - 1, 3x + 1). Find x if the circle passes through (-3, -1) and the length of its diameter is 20 unit.

The length of line PQ is 10 units and the co-ordinates of P are (2, -3); calculate the co-ordinates of point Q, if its abscissa is 10.

Point P (2, -7) is the center of a circle with radius 13 unit, PT is perpendicular to chord AB and T = (-2, -4); calculate the length of: AT 

Point P (2, -7) is the centre of a circle with radius 13 unit, PT is perpendicular to chord AB and T = (-2, -4); calculate the length of AB.

Calculate the distance between the points P (2, 2) and Q (5, 4) correct to three significant figures.

Calculate the distance between A (7, 3) and B on the x-axis, whose abscissa is 11.

Calculate the distance between A (5, -3) and B on the y-axis whose ordinate is 9.

Find the point on y-axis whose distances from the points A (6, 7) and B (4, -3) are in the ratio 1: 2.

The distances of point P (x, y) from the points A (1, - 3) and B (- 2, 2) are in the ratio 2: 3.
Show that: 5x² + 5y² - 34x + 70y + 58 = 0.

The points A (3, 0), B (a, -2) and C (4, -1) are the vertices of triangle ABC right angled at vertex A. Find the value of a.

If two vertices of an equilateral triangle be (0, 0), (3, `sqrt3`), find the third vertex using distance formula.

A group of class IX students goes to a picnic during winter holidays. The positions of three friends Nitin, Rajesh and Kareem are shown by the points P, Q and R respectively.

1. Find the distance between:
   1. Nitin and Rajesh
   2. Rajesh and Kareem
   3. Nitin and Kareem
2. Show that P, Q and R are collinear.
3. Find the point on X-axis which is equidistant from points Q and R.
4. If R is taken as origin; what will be the coordinates of P and Q?

Rohan and Sohan stay in a village which are 50 km apart and are situated on Delhi Agra Highway as shown in the figure. Another highway YY’ crosses Agra Delhi highway at O (0, 0). A small local road PQ crosses both the highways at points A and B such that OA = 10 km and OB = 12 km. Also the villages of Sita and Gita are on the smaller highway YY’. Sita’s village is 12 km from O and Gita’s village is 15 km from O.

Now answer the following questions:

1. Find the Coordinates of A and B.
2. Find the Length of local road AB.
3. What is the shortest distance Sohan will cover to reach Sita’s village.
4. How far is Rohan’s village from Gita’s village?