Co­ordinate Geometry Exercise 14.1 (Part-7) | Extra Documents, Videos & Tests for Class 10 PDF Download

Question 21: Find the ratio in which the point P(−1, y) lying on the line segment joining A(−3, 10) and B(6 −8) divides it. Also find the value of y.   

Answer : Suppose P(−1, y) divides the line segment joining A(−3, 10) and B(6 −8) in the ratio k : 1. 

Using section formula, we get 

Coordinates of P = 

Now

, 

So, P divides the line segment AB in the ratio 2 : 7. 

Putting k = 2/7 in Y =

Hence, the value of y is 6. 

Question 22: Find the coordinates of a point A, where AB is a diameter of the circle whose centre is (2, −3) and Bis (1, 4).

Answer :

Let the co-ordinates of point A be.

Centre lies on the mid-point of the diameter. So applying the mid-point formula we get,

x = 3

Similarly, 

y = -10

So the co-ordinates of A are (3,−10) 

Question 23: If the points (−2, −1), (1, 0), (x, 3) and  (1, y) form a parallelogram, find the values of x and y.

Answer : 

Let ABCD be a parallelogram in which the co-ordinates of the vertices are A (−2,−1); B (1, 0); C (x, 3) and D (1, y).

Since ABCD is a parallelogram, the diagonals bisect each other. Therefore the mid-point of the diagonals of the parallelogram will coincide.

In general to find the mid-point of two pointsand we use section formula as,

The mid-point of the diagonals of the parallelogram will coincide.

So,

Therefore,

Now equate the individual terms to get the unknown value. So,

Similarly,

Therefore,

Question 24: The points A(2, 0), B(9, 1) C(11, 6) and D(4, 4) are the vertices of a quadrilateral ABCD. Determine whether ABCD is a rhombus or not.

Answer :

Let A (2, 0); B (9, 1); C (11, 6) and D (4, 4) be the vertices of a quadrilateral. We have to check if the quadrilateral ABCD is a rhombus or not.

So we should find the lengths of sides of quadrilateral ABCD.

All the sides of quadrilateral are unequal. Hence ABCD is not a rhombus.

Question 25: In what ratio does the point (−4, 6) divide the line segment joining the points A(−6, 10) and B(3,−8)?

Answer :

The co-ordinates of a point which divided two points  and  internally in the ratio is given by the formula,

Here it is said that the point (−4,6) divides the points A(−6,10) and B(3,−8). Substituting these values in the above formula we have,

Equating the individual components we have,

Therefore the ratio in which the line is divided is

Question 26: Find the ratio in which the y-axis divides the line segment joining the points (5, −6)  and (−1,−4). Also, find the coordinates of the point of division.

Answer  : 

The ratio in which the y-axis divides two points  and  is λ:1λ:1

The co-ordinates of the point dividing two points  and  in the ratio  is given as,

 where, 

Here the two given points are A(5,−6) and B(−1,−4).

Since, the y-axis divided the given line, so the x coordinate will be 0. 

Thus the given points are divided by the y-axis in the ratio.

The co-ordinates of this point (x, y) can be found by using the earlier mentioned formula.

Thus the co-ordinates of the point which divides the given points in the required ratio are.

Question 27: Show that A (−3, 2), B (−5, −5), C (2,−3), and D (4, 4) are the vertices of a rhombus.

Answer :

Let A (−3, 2); B (−5,−5); C (2,−3) and D (4, 4) be the vertices of a quadrilateral. We have to prove that the quadrilateral ABCD is a rhombus.

So we should find the lengths of sides of quadrilateral ABCD.

All the sides of quadrilateral are equal. Hence ABCD is a rhombus.

Question 28: Find the length of the medians of a ΔABC having vertices at A(0, −1), B(2, 1) and C(0, 3).

Answer :

We have to find the lengths of the medians of a triangle whose co-ordinates of the vertices are A (0,−1); B (2, 1) and C (0, 3).

So we should find the mid-points of the sides of the triangle.

In general to find the mid-point of two pointsand we use section formula as,

Therefore mid-point P of side AB can be written as,

Now equate the individual terms to get,

So co-ordinates of P is (1, 0)

Similarly mid-point Q of side BC can be written as,

Now equate the individual terms to get,

So co-ordinates of Q is (1, 2)

Similarly mid-point R of side AC can be written as,

Now equate the individual terms to get,

So co-ordinates of R is (0, 1)

Therefore length of median from A to the side BC is,

Similarly length of median from B to the side AC is,

Similarly length of median from C to the side AB is

Question 29: Find the lengths of the medians of a ΔABC having vertices at A(5, 1), B(1, 5), and C(−3, −1).

Answer : 

We have to find the lengths of the medians of a triangle whose co-ordinates of the vertices are A (5, 1); B (1, 5) and C (−3,−1).

So we should find the mid-points of the sides of the triangle.

In general to find the mid-point of two pointsand we use section formula as,

Therefore mid-point P of side AB can be written as,

Now equate the individual terms to get,

So co-ordinates of P is (3, 3)

Similarly mid-point Q of side BC can be written as,

Now equate the individual terms to get,

So co-ordinates of Q is (−1, 2)

Similarly mid-point R of side AC can be written as,

Now equate the individual terms to get,

So co-ordinates of R is (1, 0)

Therefore length of median from A to the side BC is,

Similarly length of median from B to the side AC is,

Similarly length of median from C to the side AB is