Question

Prove that the straight line given by the equation
(3x+4y+7)+k(5x-2y-7)=0
passes through a fixed point for all value of k and find the point

Answer 1

@hartnn please help :P

Answer 2

find out the intersection of two lines 3x+4y+7=0 and 5x-2y-7=0

Answer 3

(3x+4y+7)+k(5x-2y-7)=0 is equation of line which passes through the intersecting points of two straight lines
3x+4y+7=0 and 5x-2y-7=0

Answer 4

does it make sense?

Answer 5

what after finding the intersection points?
I didn't get it?

Answer 6

(3x+4y+7)+k(5x-2y-7)=0
this is true for all k...right?
so it's an identity in k that means k has multiple roots.
so equate the coefficient of k and constant term equal to zero

Answer 7

so 3x+4y+7=0
and 5x-2y-7=0 then only (3x+4y+7)+k(5x-2y-7)=0 for all k

Answer 8

now solve 3x+4y+7=0
and 5x-2y-7=0
find out the values of x and y

Answer 9

x=0 and y= -7/4

Answer 10

so for x=0 and y=-7/4
(3x+4y+7)+k(5x-2y-7)=0 for all k...right?

Answer 11

yeah so?

Answer 12

x=0 and y=-7/4 is that point

Answer 13

that's not the answer

Answer 14

then u didn't solve the equations correctly...

Answer 15

answer is (3,4)

Answer 16

(3x+4y+7)+k(5x-2y-7)=0 ...have u written it correctly?

Answer 17

yeah

Answer 18

(3,4) won't satisfy
3x+4y+7=0
3x+4y+7=0...so there must be some typo in the book

Answer 19

5x-2y-7=0

Answer 20

but did u get the solution?

Answer 21

suppose u have to write down the equation of straight line passing through the intersection point of 3x+4y+7=0 and 5x-2y-7=0?

Answer 22

I still didn't understand it...why are we taking the equations differently?

Answer 23

just wait...suppose u have been given two equations 3x+4y+7=0 and 5x-2y-7=0 and u have been asked to write down the equation of a straight line, which passes through the intersection of given two lines.

Answer 24

right?

Answer 25

yeah and?

Answer 26

so equation of straight line passing through P will be
(3x+4y+7)+k(5x-2y-7)=0
check whether coordinate of P will satisfy this equation or not?

Answer 27

okkk

Answer 28

or other way..
(3x+4y+7)+k(5x-2y-7)=0
this is true for all k...right?
so it's an identity in k that means k has multiple roots.
right?

Answer 29

yep

Answer 30

i mean ...this is a linear equation in k...right?

Answer 31

yeah

Answer 32

so this equation has one root?

Answer 33

for k

Answer 34

one root?how?

Answer 35

2x+3=0...how many root?

Answer 36

2k+3=0

Answer 37

one

Answer 38

yes..then (3x+4y+7)+k(5x-2y-7)=0 is a linear equation in k...right?

Answer 39

since k= -3/2

Answer 40

yes

Answer 41

yes

Answer 42

but u r being told in the question that
(3x+4y+7)+k(5x-2y-7)=0 is true for all k..it means this equation has multiple roots for k

Answer 43

yeah now?

Answer 44

right?

Answer 45

so this is an identity in k

Answer 46

i see

Answer 47

so was 0 and -7/4 right answers?

Answer 48

suppose
a + b k=0 and this is true for all k...then a=b=0...right?

Answer 49

k and?>

Answer 50

similarly (3x+4y+7)+k(5x-2y-7)=0 then
3x+4y+7=0 and 5x-2y-7=0

Answer 51

did u get it?

Answer 52

i did

Answer 53

ok..so if (3x+4y+7)+k(5x-2y-7)=0 then
3x+4y+7=0 and 5x-2y-7=0

that means (3x+4y+7)+k(5x-2y-7)=0 will pass through the intersecting point of 3x+4y+7=0 and 5x-2y-7=0

Answer 54

yeah.....

Answer 55

so now u got it...answer might be wrong...but i think u didnt solve the equations correctly

Answer 56

as u got x=0, y=-7/4

Answer 57

what are you getting?

Answer 58

does it satisfy the equation
5x-2y-7=0

Answer 59

i didnt solve...but i can tell it's wrong

Answer 60

i did it twice and I got the same answer...I used the elimination method

Answer 61

x=7/13 and y= -112/52

Answer 62

how you did it? Elimination method?

Answer 63

3x+4y+7=0 and 5x-2y-7=0
multiply the 2nd equation by 2 and add it to the 1st equation

Answer 64

u'll get 13x=7

Answer 65

yeah i did that

Answer 66

so x=13/7

Answer 67

fine?