The distance between two points, P ( a , - 1 ) and Q ( - 5 , a ) is 4 5 . Find the two possible values of a .

Distance between two points x 1 , y 1 and x 2 , y 2 is x 2 - x 1 2 + y 2 - y 1 2 Distance of P Q = ( - 5 - a ) 2 + ( a - ( - 1 ) ) 2 = 4 5 Squaring on both sides ⇒ a 2 + 25 + 10 a + a 2 + 1 + 2 a = 80 ⇒ 2 a 2 + 12 a - 54 = 0 ⇒ a 2 + 6 a - 27 = 0 ⇒ a 2 + 9 a - 3 a - 27 = 0 ⇒ a ( a + 9 ) - 3 ( a + 9 ) = 0 ⇒ ( a + 9 ) ( a - 3 ) = 0 ∴ Possible values of a are 3 , - 9 .

The distance between two points, P ( - 3 , - 2 ) and Q ( b , 2 b ) is 10 . Find the two possible values of b .

Distance between two points x 1 , y 1 and x 2 , y 2 is x 2 - x 1 2 + y 2 - y 1 2 Distance of P Q = ( b + 3 ) 2 + ( 2 b + 2 ) 2 = 10 Squaring on both sides ⇒ b 2 + 9 + 6 b + 4 b 2 + 4 + 8 b = 100 ⇒ 5 b 2 + 14 b - 87 = 0 ⇒ 5 b 2 - 15 b + 29 b - 87 = 0 ⇒ 5 b ( b - 3 ) + 29 ( b - 3 ) = 0 ⇒ ( 5 b + 29 ) ( b - 3 ) = 0 ∴ Possible values of b are 3 , - 29 5 .

The line x + y = 4 meets the curve y = 8 - 5 x at the points A and B . Find the coordinates of the midpoint of A B .

Given line is x + y = 4 and curve is y = 8 - 5 x Equation of curve ⇒ x y = 8 x - 5 For intersection points put line equation in curve. ⇒ x ( 4 - x ) = 8 x - 5 ⇒ 4 x - x 2 = 8 x - 5 ⇒ x 2 + 4 x - 5 = 0 ⇒ ( x - 1 ) ( x + 5 ) = 0 ⇒ x = 1 , - 5 Similarly put x = 4 - y in curve. ⇒ y ( 4 - y ) = 8 ( 4 - y ) - 5 ⇒ 4 y - y 2 = 32 - 8 y - 5 ⇒ y 2 - 12 y + 27 = 0 ⇒ ( y - 9 ) ( y - 3 ) = 0 ⇒ y = 3 , 9 Let A = ( 1 , 3 ) and B = ( - 5 , 9 ) ∴ Midpoint of A B = 1 - 5 2 , 3 + 9 2 = ( - 2 , 6 )

E M B I B E

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 3 - Coordinate Geometry>EXERCISE 3A

Sue Pemberton Solutions for Chapter: Coordinate Geometry, Exercise 2: EXERCISE 3A

Author:Sue Pemberton

Sue Pemberton Mathematics Solutions for Exercise - Sue Pemberton Solutions for Chapter: Coordinate Geometry, Exercise 2: EXERCISE 3A

Attempt the practice questions on Chapter 3: Coordinate Geometry, Exercise 2: EXERCISE 3A with hints and solutions to strengthen your understanding. Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book solutions are prepared by Experienced Embibe Experts.

Questions from Sue Pemberton Solutions for Chapter: Coordinate Geometry, Exercise 2: EXERCISE 3A with Hints & Solutions

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 3 - Coordinate Geometry>EXERCISE 3A>Q 3

The distance between two points, $P (a, - 1)$ and $Q (- 5, a)$ is $4 \sqrt{5}$ . Find the two possible values of $a$ .

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 3 - Coordinate Geometry>EXERCISE 3A>Q 4

The distance between two points, $P (- 3, - 2)$ and $3, - 9$ is $10$ . Find the two possible values of $b$ .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 3 - Coordinate Geometry>EXERCISE 3A>Q 5

The point $(- 2, - 3)$ is the midpoint of the line segment joining $P (- 6, - 5)$ and $Q (a, b)$ . Find the value of $a$ and the value of $b$ .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 3 - Coordinate Geometry>EXERCISE 3A>Q 7

The point $P (k, 2 k)$ is equidistant from $A (8, 11)$ and $B (1, 12)$ . Find the value of $k$ .

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 3 - Coordinate Geometry>EXERCISE 3A>Q 8

Triangle $A B C$ has vertices at $A (- 6, 3), B (3, 5)$ and $C (1, - 4)$ . Show that triangle $A B C$ is isosceles and find the area of this triangle.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 3 - Coordinate Geometry>EXERCISE 3A>Q 9

Triangle $A B C$ has vertices at $A (- 7, 8), B (3, k)$ and $C (8, 5)$ . Given that $A B = 2 B C$ , find the value of $k$ .

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 3 - Coordinate Geometry>EXERCISE 3A>Q 10

The line $x + y = 4$ meets the curve $y = 8 - \frac{5}{x}$ at the points $A$ and $B$ . Find the coordinates of the midpoint of $A B$ .

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 3 - Coordinate Geometry>EXERCISE 3A>Q 12

In triangle $A B C$ , the midpoints of the sides $A B, B C$ and $A C$ are $(1, 4), (2, 0)$ and $(- 4, 1)$ , respectively. Find the coordinates of points $A, B$ and $C$ .

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