# Surds & Length of straight line

• synkk
In summary, the normal to C at P (4,8) cuts the x-axis at the point Q. To find the length PQ of the point Q, you would need to solve for x using the equation of the normal, and then simplify the result.

## Homework Statement

1)The normal to C ( 3y = x + 20 ) at P (4, 8) cuts the x-axis at the point Q.
Find the length PQ, giving your answers in a simplified surd form.

2) write $$\dfrac{2\sqrt{x} + 3}{x}$$ in the form $$2x^p + 3x^q$$ where p and q are constants

## Homework Equations

y = mx + c? I'm not sure

## The Attempt at a Solution

for 1) i let y = 0 and got Q (-20,0) but i don't know how to find the length of the two points.

for 2)
$$\dfrac{3}{x} = 3x^{-1}$$ but i don't know what $$\dfrac{2\sqrt{x}}{x}$$ is.

thanks

Last edited:
synkk said:

## Homework Statement

1)The normal to C ( 3y = x + 20 ) at P (4, 8) cuts the x-axis at the point Q.
Find the length PQ, giving your answers in a simplified surd form.

## The Attempt at a Solution

for 1) i let y = 0 and got Q (-20,0) but i don't know how to find the length of the two points.

Q is the point where the normal intersects the x axis. You have to start by writing the equation of the normal line. And you don't find the "length of two points". You find the distance between the two points.

synkk said:
2) write $$\dfrac{2\sqrt{x} + 3}{x}$$ in the form $$2x^p + 3x^q$$ where p and q are constants

$$\dfrac{3}{x} = 3x^{-1}$$ but i don't know what $$\dfrac{2\sqrt{x}}{x}$$ is.

thanks

Write it with exponents and use the rules of exponents to simplify it.

LCKurtz said:
Write it with exponents and use the rules of exponents to simplify it.

Alright, for 1)

I let y = 0 and got Q as (-20,0) (seeing as it says it crosses the x-axis)

so $$\sqrt{(4-(-20))^2 + (8-0)^2} = \sqrt{640} = 8\sqrt{10}$$
for 2)

$$\dfrac{2\sqrt{x}}{x} = 2x^{-\dfrac{1}{2}} = 2x^{-\dfrac{1}{2}} + 3x^{-1}$$

Have i gone wrong anywhere?

synkk said:
Alright, for 1)

I let y = 0 and got Q as (-20,0) (seeing as it says it crosses the x-axis)
Did you even read my post?? Q = (-20,0) has nothing to do with the problem.
for 2)

$$\dfrac{2\sqrt{x}}{x} = 2x^{-\dfrac{1}{2}} = 2x^{-\dfrac{1}{2}} + 3x^{-1}$$

You have worked the exponents correctly, but that last line has = signs between things that aren't equal.

LCKurtz said:
Did you even read my post?? Q = (-20,0) has nothing to do with the problem.

You have worked the exponents correctly, but that last line has = signs between things that aren't equal.
I wrote = just to show it goes to that answer, but whatever i understand that question now, thanks.

For 1) the question says that the normal to C at P is 3y = x + 20 (its on the question before that), P is (4,8) also on the question before that.

It says that the normal at (4,8) cuts the x-axis at Q, to find where it crosses the x-axis you would have to let y = 0 no? If not I'm completely mistaken as that's the only way I'm seeing the question at the moment, i don't see what your trying to say.

thanks again.

synkk said:

## Homework Statement

1)The normal to C ( 3y = x + 20 ) at P (4, 8) cuts the x-axis at the point Q.

synkk said:
For 1) the question says that the normal to C at P is 3y = x + 20 (its on the question before that), P is (4,8) also on the question before that.

We aren't mind readers here. The statement I have bolded is readily interpreted "The normal to [the curve] C [whose equation is] 3y = x + 20 ... meaning you have this curve C whose equation is given and the first problem is to find its normal. Why do you even mention it is normal to some curve C which isn't given and is irrelevant? How are we supposed to know what is on "the question before that"?

LCKurtz said:
We aren't mind readers here. The statement I have bolded is readily interpreted "The normal to [the curve] C [whose equation is] 3y = x + 20 ... meaning you have this curve C whose equation is given and the first problem is to find its normal. Why do you even mention it is normal to some curve C which isn't given and is irrelevant? How are we supposed to know what is on "the question before that"?

I thought it was pretty clear that the NORMAL TO C (3y=x+20), i didn't want to mention other questions because I've already solved the other questions and mentioned the data which was needed for this question...

thanks for your help anyway, appreciated.