**Chapter 5 Co-ordinate Geometry Set 5**

**Chapter 5 Co-ordinate Geometry Set 5**

**Question 1.Fill in the blanks using correct alternatives.**

**i. Seg AB is parallel to Y-axis and co-ordinates of point A are (1, 3), then co-ordinates of point B can be _______.(A) (3,1)(B) (5,3)(C) (3,0)(D) (1,-3)Answer: **(D)

Since, seg AB || Y-axis.

âˆ´ x co-ordinate of all points on seg AB

will be the same,

x co-ordinate of A (1, 3) = 1

x co-ordinate of B (1, â€“ 3) = 1

âˆ´ Option (D) is correct.

**ii. Out of the following, point lies to the right of the origin on X-axis.(A) (-2,0)(B) (0,2)(C) (2,3)(D) (2,0)Answer: **(D)

**iii. Distance of point (-3, 4) from the origin is _________.(A) 7(B) 1(C) 5(D) -5Answer: **(C)

Distance of (-3, 4) from origin

**iv. A line makes an angle of 30Â° with the positive direction of X-axis. So the slope of the line is ________.Answer: **(C)

**Question 2.Determine whether the given points are collinear.i. A (0, 2), B (1, -0.5), C (2, -3)ii. P(1,2), Q(2,85),R(3,65)iii L (1, 2), M (5, 3), N (8, 6)Solution:**

âˆ´ slope of line AB = slope of line BC

âˆ´ line AB || line BC

Also, point B is common to both the lines.

âˆ´ Both lines are the same.

âˆ´ Points A, B and C are collinear.

âˆ´ slope of line PQ = slope of line QR

âˆ´ line PQ || line QR

Also, point Q is common to both the lines.

âˆ´ Both lines are the same.

âˆ´ Points P, Q and R are collinear.

âˆ´ slope of line LM â‰ slope of line MN

âˆ´ Points L, M and N are not collinear.

[Note: Students can solve the above problems by using distance formula.]

**Question 3.Find the co-ordinates of the midpoint of the line segment joining P (0,6) and Q (12,20).Solution:**

P(x

_{1},y

_{1}) = P (0, 6), Q(x

_{2}, y

_{2}) = Q (12, 20)

Here, x

_{1}= 0, y

_{1}= 6, x

_{2}= 12, y

_{2}= 20

âˆ´ Co-ordinates of the midpoint of seg PQ

âˆ´ The co-ordinates of the midpoint of seg PQ are (6,13).

**Question 4.Find the ratio in which the line segment joining the points A (3, 8) and B (-9, 3) is divided by the Y-axis.Solution:**

Let C be a point on Y-axis which divides seg AB in the ratio m : n.

Point C lies on the Y-axis

âˆ´ its x co-ordinate is 0.

Let C = (0, y)

Here A (x

_{1},y

_{1}) = A(3, 8)

B (x

_{2}, y

_{2}) = B (-9, 3)

âˆ´ By section formula,

âˆ´ Y-axis divides the seg AB in the ratio 1 : 3.

**Question 5.Find the point on X-axis which is equidistant from P (2, -5) and Q (-2,9).Solution:**

Let point R be on the X-axis which is equidistant from points P and Q.

Point R lies on X-axis.

âˆ´ its y co-ordinate is 0.

Let R = (x, 0)

R is equidistant from points P and Q.

âˆ´ PR = QR

âˆ´ (x â€“ 2)

^{2}+ [0 â€“ (-5)]

^{2}= [x â€“ (- 2)]

^{2}+ (0 â€“ 9)

^{2}â€¦[Squaring both sides]

âˆ´ (x â€“ 2)

^{2}+ (5)

^{2}= (x + 2)

^{2}+ (-9)

^{2}

âˆ´ 4 â€“ 4x + x

^{2}+ 25 = 4 + 4x + x

^{2}+ 81

âˆ´ â€“ 8x = 56

âˆ´ x = -7

âˆ´ The point on X-axis which is equidistant from points P and Q is (-7,0).

**Question 6.Find the distances between the following points.i. A (a, 0), B (0, a)ii. P (-6, -3), Q (-1, 9)iii. R (-3a, a), S (a, -2a)Solution:**

i. Let A (x

_{1}, y

_{1}) and B (x

_{2}, y

_{2}) be the given points.

âˆ´ x

_{1}= a, y

_{1}= 0, x

_{2}= 0, y

_{2}= a

By distance formula,

âˆ´ d(A, B) = a2â€“âˆš units

ii. Let P (x_{1}, y_{1}) and Q (x_{2}, y_{2}) be the given points.

âˆ´ x_{1} = -6, y_{1} = -3, x_{2} = -1, y_{2} = 9

By distance formula,

âˆ´ d(P, Q) = 13 units

iii. Let R (x_{1}, y_{1}) and S (x_{2}, y_{2}) be the given points.

âˆ´ x_{1} = -3a, y_{1} = a, x_{2} = a, y_{2} = -2a

By distance formula,

âˆ´ d(R, S) = 5a units

**Question 7.Find the co-ordinates of the circumcentre of a triangle whose vertices are (-3,1), (0, -2) and (1,3).Solution:**

Let A (-3, 1), B (0, -2) and C (1, 3) be the vertices of the triangle.

Suppose O (h, k) is the circumcentre of âˆ†ABC.

âˆ´ (h + 3)

^{2}+ (k â€“ 1)

^{2}= h

^{2}+ (k + 2)

^{2}

âˆ´ h

^{2}+ 6h + 9 + k

^{2}â€“ 2k + 1 = h

^{2}+ k

^{2}+ 4k + 4

âˆ´ 6h â€“ 2k + 10 = 4k + 4

âˆ´ 6h â€“ 2k â€“ 4k = 4 â€“ 10

âˆ´ 6h â€“ 6k = â€“ 6

âˆ´ h â€“ k = -1 ,..(i)[Dividing both sides by 6]

OB = OC â€¦[Radii of the same circle]

âˆ´ h

^{2}+ (k + 2)

^{2}= (h â€“ 1)

^{2}+ (k â€“ 3)

^{2}

âˆ´ h

^{2}+ k

^{2}+ 4k + 4 = h

^{2}â€“ 2h + 1 + k

^{2}â€“ 6k + 9

âˆ´ 4k + 4 = -2h + 1 â€“ 6k + 9

âˆ´ 2h+ 10k = 6

âˆ´ h + 5k = 3 â€¦(ii)

Subtracting equation (ii) from (i), we get

âˆ´ The co-ordinates of the circumcentre of the triangle are ()

**Question 8.In the following examples, can the segment joining the given points form a triangle? If triangle is formed, state the type of the triangle considering sides of the triangle.i. L (6, 4), M (-5, -3), N (-6, 8)ii. P (-2, -6), Q (-4, -2), R (-5, 0)iii. A(2â€“âˆš,2â€“âˆš),B(-2â€“âˆš,-2â€“âˆš),C(6â€“âˆš,6â€“âˆš)Solution:**

i. By distance formula,

âˆ´ d(M, N) + d (L, N) > d (L, M)

âˆ´ Points L, M, N are non collinear points.

We can construct a triangle through 3 non collinear points.

âˆ´ The segment joining the given points form a triangle.

Since MN â‰ LN â‰ LM

âˆ´ âˆ†LMN is a scalene triangle.

âˆ´ The segments joining the points L, M and N will form a scalene triangle.

ii. By distance formula,

âˆ´ d(P, Q) + d(Q, R) = d (P, R) â€¦[From (iii)]

âˆ´ Points P, Q, R are collinear points.

We cannot construct a triangle through 3 collinear points.

âˆ´ The segments joining the points P, Q and R will not form a triangle.

iii. By distance formula,

âˆ´ d(A, B) + d(B, C) + d(A, C) â€¦ [From (iii)]

âˆ´ Points A, B, C are non collinear points.

We can construct a triangle through 3 non collinear points.

âˆ´ The segment joining the given points form a triangle.

Since, AB = BC = AC

âˆ´ âˆ†ABC is an equilateral triangle.

âˆ´ The segments joining the points A, B and C will form an equilateral triangle.

**Question 9.Find k, if the line passing through points P (-12, -3) and Q (4, k) has slope .Solution:**

P(x

_{1},y

_{1}) = P(-12,-3),

Q(X

_{2},T

_{2}) = Q(4, k)

Here, x

_{1}= -12, x

_{2}= 4, y

_{1}= -3, y

_{2}= k

**Question 10.Show that the line joining the points A (4,8) and B (5, 5) is parallel to the line joining the points C (2, 4) and D (1 ,7).Proof:**
âˆ´ Slope of line AB = Slope of line CD

Parallel lines have equal slope.

âˆ´ line AB || line CD

**Question 11.Show that points P (1, -2), Q (5, 2), R (3, -1), S (-1, -5) are the vertices of a parallelogram.Proof:**

By distance formula,

In ê ¸PQRS,

PQ = RS â€¦ [From (i) and (iii)]

QR = PS â€¦ [From (ii) and (iv)]

âˆ´ ê ¸ PQRS is a parallelogram.

[A quadrilateral is a parallelogram, if both the pairs of its opposite sides are congruent]

âˆ´ Points P, Q, R and S are the vertices of a parallelogram.

**Question 12.Show that the ê ¸PQRS formed by P (2, 1), Q (-1, 3), R (-5, -3) and S (-2, -5) is a rectangle.Proof:**
By distance formula,

In ê ¸PQRS,

PQ = RS â€¦[From (i) and (iii)]

QR = PS â€¦[From (ii) and (iv)]

ê ¸PQRS is a parallelogram.

[A quadrilateral is a parallelogram, if both the pairs of its opposite sides are congruent]

In parallelogram PQRS,

PR = QS â€¦ [From (v) and (vi)]

âˆ´ ê ¸PQRS is a rectangle.

[A parallelogram is a rectangle if its diagonals are equal]

**Question 13.Find the lengths of the medians of a triangle whose vertices are A (-1, 1), B (5, -3) and C (3,5).Solution:**

Suppose AD, BE and CF are the medians.

âˆ´ Points D, E and F are the midpoints of sides BC, AC and AB respectively.

âˆ´ By midpoint formula,

**Question 14.Find the co-ordinates of centroid of the triangle if points D (-7, 6), E (8, 5) and F (2, -2) are the mid points of the sides of that triangle.Solution:**

Suppose A (x

_{1}, y

_{1}), B (x

_{2}, y

_{2}) and C (x

_{3}, y

_{3}) are the vertices of the triangle.

D (-7, 6), E (8, 5) and F (2, -2) are the midpoints of sides BC, AC and AB respectively.

Let G be the centroid of âˆ†ABC.

D is the midpoint of seg BC.

By midpoint formula,

E is the midpoint of seg AC.

By midpoint formula,

Adding (i), (iii) and (v),

x

_{2}+ x

_{3}+ x

_{1}+ x

_{3}+ x

_{1}+ x

_{2}= -14 + 16 + 4

âˆ´ 2x

_{1}+ 2x

_{2}+ 2x

_{3}= 6

âˆ´ x

_{1}+ x

_{2}+ x

_{3}= 3 â€¦(vii)

Adding (ii), (iv) and (vi),

y

_{2}+ y

_{3}+ y

_{1}+ y

_{3}+ y

_{1}+y

_{2}= 12 + 10 â€“ 4

âˆ´ 2y

_{1}+ 2y

_{2}+ 2y

_{3}= 18

âˆ´ y

_{1}+ y

_{2}+ y

_{3}= 9 â€¦(viii)

G is the centroid of âˆ†ABC.

By centroid formula,

âˆ´ The co-ordinates of the centroid of the triangle are (1,3).

**Question 15.Show that A (4, -1), B (6, 0), C (7, -2) and D (5, -3) are vertices of a square.Proof:**

By distance formula,

âˆ´ â–¡ABCD is a square.

[A rhombus is a square if its diagonals are equal]

**Question 16.Find the co-ordinates of circumcentre and radius of circumcircle of AABC if A (7, 1), B (3,5) and C (2,0) are given.Solution:**

Suppose, O (h, k) is the circumcentre of âˆ†ABC

âˆ´ h

^{2}â€“ 6h + 9 + k

^{2}â€“ 10k + 25 = h

^{2}â€“ 4h + 4 + k

^{2}

âˆ´ 2h + 10k = 30

âˆ´ h + 5k = 15 â€¦ (ii)[Dividing both sides by 2]

Multiplying equation (i) by 5, we get

25h + 5k = 115 â€¦(iii)

Subtracting equation (ii) from (iii), we get

Substituting the value of h in equation (i), we get

**Question 17.Given A (4, -3), B (8, 5). Find the co-ordinates of the point that divides segment AB in the ratio 3:1.Solution:**

Suppose point C divides seg AB in the ratio 3:1.

Here; A(x

_{1}, y

_{1}) = A (4, -3)

B (x

_{2}, y

_{2}) = B (8, 5)

By section formula,

âˆ´ The co-ordinates of point dividing seg AB in ratio 3 : 1 are (7, 3).

**Question 18.Find the type of the quadrilateral if points A (-4, -2), B (-3, -7), C (3, -2) and D (2, 3) are joined serially.Solution:**

Slope of AB = slope of CD

âˆ´ line AB || line CD

slope of BC = slope of AD

âˆ´ line BC || line AD

Both the pairs of opposite sides of âˆ†ABCD are parallel.

âˆ´ ê ¸ ABCD is a parallelogram.

âˆ´ The quadrilateral formed by joining the points A, B, C and D is a parallelogram.

**Question 19.The line segment AB is divided into five congruent parts at P, Q, R and S such that A-P-Q-R-S-B. If point Q (12, 14) and S (4, 18) are given, find the co-ordinates of A, P, R, B.Solution:**

Points P, Q, R and S divide seg AB in five congruent parts.

Let A (x

_{1}, y

_{1}), B (x

_{2}, y

_{2}), P (x

_{3}, y

_{3}) and

R (x

_{4}, y

_{4}) be the given points.

Point R is the midpoint of seg QS.

By midpoint formula,

âˆ´ co-ordinates of R are (8, 16).

Point Q is the midpoint of seg PR.

By midpoint formula,

âˆ´ 28 = y

_{3}+ 16

âˆ´ y

_{3}= 12

âˆ´ P(x

_{3},y

_{3}) = (16, 12)

âˆ´ co-ordinates of P are (16, 12).

Point P is the midpoint of seg AQ.

By midpoint formula,

âˆ´ co-ordinates of A are (20, 10).

Point S is the midpoint of seg RB.

By midpoint formula,

âˆ´ 36 = y

_{2}+ 16

âˆ´ y

_{2}= 20

âˆ´ B(x

_{2}, y

_{2}) = (0, 20)

âˆ´ co-ordinates of B are (0, 20).

âˆ´ The co-ordinates of points A, P, R and B are (20, 10), (16, 12), (8, 16) and (0, 20) respectively.

**Question 20.Find the co-ordinates of the centre of the circle passing through the points P (6, -6), Q (3, -7) and R (3,3).Solution:**

Suppose O (h, k) is the centre of the circle passing through the points P, Q and R.

âˆ´ (h â€“ 6)

^{2}+ (k + 6)

^{2}= (h â€“ 3)

^{2}+ (k + 7)

^{2}

âˆ´ h

^{2}â€“ 12h + 36 + k

^{2}+ 12k + 36

= h

^{2}â€“ 6h + 9 + k

^{2}+ 14k + 49

âˆ´ 6h + 2k = 14

âˆ´ 3h + k = 7 â€¦(i)[Dividing both sides by 2]

OP = OR â€¦[Radii of the same circle]

âˆ´ (h â€“ 6)

^{2}+ (k + 6)

^{2}= (h â€“ 3)

^{2}+ (k â€“ 3)

^{2}

âˆ´ h

^{2}â€“ 12h + 36 + k

^{2}+ 12k + 36

= h

^{2}â€“ 6h + 9 + k

^{2}â€“ 6k + 9

âˆ´ 6h â€“ 18k = 54

âˆ´ 3h â€“ 9k = 27 â€¦(ii)[Dividing both sides by 2]

Subtracting equation (ii) from (i), we get

Substituting the value of k in equation (i), we get

3h â€“ 2 = 7

âˆ´ 3h = 9

âˆ´ h = 93 = 3

âˆ´ The co-ordinates of the centre of the circle are (3, -2).

**Question 21.Find the possible pairs of co-ordinates of the fourth vertex D of the parallelogram, if three of its vertices are A (5, 6), B (1, -2) and C (3, -2).Solution:**

Let the points A (5, 6), B (1, -2) and C (3, -2) be the three vertices of a parallelogram.

The fourth vertex can be point D or point Di or point D

_{2}as shown in the figure.

Let D(x

_{1},y

_{1}), D, (x

_{2}, y

_{2}) and D

_{2}(x

_{3},y

_{3}).

Consider the parallelogram ACBD.

The diagonals of a parallelogram bisect each other.

âˆ´ midpoint of DC = midpoint of AB

Co-ordinates of point D(x

_{1}, y

_{1}) are (3, 6).

Consider the parallelogram ABD

_{1}C.

The diagonals of a parallelogram bisect each other.

âˆ´ midpoint of AD

_{1}= midpoint of BC

âˆ´ Co-ordinates of D

_{1}(x

_{2},y

_{2}) are (-1,-10).

Consider the parallelogram ABCD

_{2}.

The diagonals of a parallelogram bisect each other.

âˆ´ midpoint of BD

_{2}= midpoint of AC

âˆ´ co-ordinates of point D

_{2}(x

_{3}, y

_{3}) are (7, 6).

âˆ´ The possible pairs of co-ordinates of the fourth vertex D of the parallelogram are (3, 6), (-1,-10) and (7,6).

**Question 22.Find the slope of the diagonals of a quadrilateral with vertices A (1, 7), B (6,3), C (0, -3) and D (-3,3).Solution:**

Suppose ABCD is the given quadrilateral.

âˆ´ The slopes of the diagonals of the quadrilateral are 10 and 0.