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**Q1. Find a point on the y-axis equidistant from (âˆ’ 5, 2) and (9, âˆ’ 2).**

**Sol. **Let the required point on the y-axis be P (0, y)

âˆ´ PA = PB

â‡’ y^{2} âˆ’ y^{2} âˆ’ 4y âˆ’ 4y = 81 + 4 âˆ’ 4 âˆ’ 25

â‡’ âˆ’ 8y = 85 âˆ’ 29

â‡’ âˆ’ 8y = 56

â‡’

âˆ´ The required point is (0, âˆ’7).

**Q2. Find a point on x-axis at a distance of 4 units from the point A (2, 1).**

**Sol. **Let the required point on x-axis be P (x, 0).

âˆ´ PA = 4

â‡’ x^{2} âˆ’ 4x + 4 + 1 = 4^{2} = 16

â‡’ x^{2} âˆ’ 4x + 1 + 4 âˆ’ 16 = 0

â‡’ x^{2} âˆ’ 4x âˆ’ 11 = 0

â‡’

**Q3. Find the distance of the point (3, âˆ’ 4) from the origin.**

**Sol. **The coordinates of origin (0, 0).

âˆ´ Distance of (3, âˆ’ 4) from the origin

**Q4. For what value of x is the distance between the points A (âˆ’ 3, 2) and B (x, 10) 10 units?**

**Sol. **The distance between A (âˆ’ 3, 2) and B (x, 10)

â‡’ (x + 3)^{2} + (8)^{2} = 10^{2}

â‡’ (x + 3)^{2} = 10^{2} âˆ’ 8^{2}

â‡’ (x + 3)^{2} = (10 âˆ’ 8) (10 + 8) = 36

â‡’

For +ve sign, x =6 âˆ’ 3 = 3

For âˆ’ve sign, x = âˆ’ 6 âˆ’ 3 = âˆ’ 9

**Q5. Find a point on the x-axis which is equidistant from the points A (5, 2) and B (1, âˆ’ 2). Sol. **The given points are: A (5, 2) and B (1, âˆ’ 2) Let the required point on the x-axis be C (x, 0).

Since, C is equidistant from A and B.

âˆ´ AC = BC

âˆ´ The required point is (0, 3).**Q6. Establish the relation between x and y when P (x, y) is equidistant from the points A (âˆ’ 1, 2) and B (2, âˆ’ 1).**

**Sol. **âˆµ P is equidistant from A and B

âˆ´ PA = PB

which is the required relation.

**Q7. Find a relation between x and y such that the point P (x, y) is equidistant from the points A (âˆ’5, 3) and B (7, 2) **

**Sol.** Since, P (x, y) is equidistant from A (âˆ’5, 3) and B (7, 2)

âˆ´ AP = BP

**Q8. Show that the points (7, âˆ’ 2), (2, 3) and (âˆ’ 1, 6) are collinear.**

**Sol. **Here, the vertices of a triangle are (7, âˆ’ 2), (2, 3) and (âˆ’ 1, 6)

âˆ´ Area of the triangle

Since area of triangle = 0

âˆ´ The vertices of the triangle are collinear.

Thus, the given points are collinear.

**Q9. Find the distance between the points **

**Sol.** Distance between is given by

**Q10. If the mid point of the line joining the points P (6, b âˆ’ 2) and Q (âˆ’ 2, 4) is (2, âˆ’ 3), find the value of b.**

**Sol. Here, P (6, b âˆ’ 2) and Q (âˆ’ 2, 4) are the given points. âˆ´ Mid point of PQ is given by:**

**Q11. In the given figure, ABC is a triangle. D and E are the mid points of the sides BC and AC respectively. Find the length of DE. Prove that **

**Sol.** Coordinates of the mid point of BC are:

Coordinates of the mid point of AC are:

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