Question

A point P whose x-coordinates is a is taken on the line y=3x-7. If Q is the point(4,1)
show that PQ2 = 10a2-56a+80. Find the value of a which will
make the expression a minimum. Hence show that the coordinates of N, the foot of the perpendicular from Q to the line are (24/5 , 1 2/5). Find the equation of QN.

I did the first part I think a is 2.8 but it's the last part I can't do. Please help!

Answer 1

Is this --> PQ2 supposed to be (PQ) ^2 ? @lionely
And, 10a2 --> supposed to be 10 a ^ 2 ?

Answer 2

Yes

Answer 3

sorry I didn't notice

Answer 4

Do you agree with the following:

Point P has coordinates (a, 3a - 7)
Point Q has coordinates (4,1)

(PQ)^2 = (4-a)^2 + [ 1 - (3a-7) ] ^ 2

Answer 5

Yes

Answer 6

PQ^2 = 10a^2 - 56a +80

dy/dx = 20a -56

a= 2.8

Answer 7

sorry for awfully late reply I'm still trying to attempt other questions.

Answer 8

I'm still here.

Answer 9

@lionely I think you may have to use implicit differentiation for the equation.

(PQ)^2 = 10a^2 - 56a +80
can be thought of as y^2 = 10x^2 - 56x + 80 where y = PQ and a = x.

y^2 = 10x^2 - 56x + 80
y = sqrt(10x^2 - 56x + 80) or y = - sqrt(10x^2 - 56x + 80)

So, y' does not appear to equal dy/dx = 20a -56

http://www.wolframalpha.com/input/?i=derivative+of+y+%3D+sqrt%2810x%5E2+-+56x+%2B+80%29

I continue to think that the value of a is incorrect and that is the impasse to solving the second portion of the problem.

Answer 10

oh

Answer 11

So I have to square root it and then differentiate?

Answer 12

In my book it's written PQ^2 the whole thing isn't squared.

Answer 13

you had to minimize the expression 10a2-56a+80
you have correctly minimized it to get a= 2.8
now, what you minimized was the distance from a point to the line, which is also 'Perpendicular Distance' geometrically.
so, the co-ordinates you get by minimizing the distance will be co-ordinates of N only.
so, x-co-ordinate of N = a = 2.8 = 2 4/5 [This confirms that your value of a is indeed correct!]
y-coordinate = 3a-7 [because N lies on that line] = 7/5 = 1 2/5
so, we proved N = (2 4/5 , 1 2/5)
now you have both points Q and N, do you know how to find equation of line given 2 points on it ??

Answer 14

Yes

Answer 15

the equation is

Answer 16

also, did you understand my explanation ? ask if any doubts....

Answer 17

3y+x-7

Answer 18

No I understood these questions are difficult to me I'm weak in geometry.

Answer 19

Thanks!

Answer 20

did you mean 3y+x=7 ?

Answer 21

because thats what i got..

Answer 22

Well isn't the same ? all I did was carry everything to one side xD

Answer 23

ohh..so you didn't mention the =0 part...
its correct 3y+x-7=0

Answer 24

oh sorry :S

Answer 25

no, problem
and welcome ^_^