f(x) and g(x) has the same gradient at point (p,q).
f(x) is a normal to the curve h(x).

Now the question: Does f(x) normal to the curve h(x) at (p,q)?

Question

f(x) and g(x) has the same gradient at point (p,q). 
f(x) is a normal to the curve h(x).

Now the question: Does f(x) normal to the curve h(x) at (p,q)?

anonymous · Answer

Original question from here, question 1 - part ii) http://dl.dropbox.com/u/63664351/STEP%201%20Final%202011.pdf Following is the solution: http://dl.dropbox.com/u/63664351/Final%20STEP%202011%20solutions.pdf Look at page 5 I quote "The gradient of the straight line is -a/b as before, so as this line is normal to the curve at thepoint (p, q)..."

anonymous · Answer

In here,$$f(x)=ax+by=1$$$$g(x)=\frac{a}{x}+\frac{b}{y}=1$$$$h(x)=\frac{a}{x}-\frac{b}{y}=1$$

anonymous · Answer

As you can read, in part (i) f(x) has the same gradient to g(x) at point (p,q).
In the question on part (ii) the question only states that f(x) is a normal to h(x) but yet in the solution, I don't know how, they just write that they are normal at (p,q).

experimentx · Answer

-a/x^2 + b/y^2 dy/dx = 0 => dy/dx = ay^2/bx^2 solve: -a/b*dy/dx = -1 find the value of dy/dx from by putting vaues of x and y here http://www.wolframalpha.com/input/?i=solve+ax%2Bby%3D1%2C+a%2Fx-b%2Fy%3D1&dataset=

experimentx · Answer

yes ... that is the condition we imposed my doing this 
m1*m2 = -1

experimentx · Answer

it think you can do long and roundabout method by the process i mentioned above.
but the solution given in your document is much shorter and better.

anonymous · Answer

I agree that $ a^2−b^2=1/2 $ But why f(x) normal with h(x) at $  (p,q) $ ?
The very point where f(x) and g(x) has the same gradient!

experimentx · Answer

it it given that point p,q lies on the line as well as the curve a/x + b/y = 1

anonymous · Answer

Yes, but in part 2 the equation is a/x - b/y =1 There is no mention whatsoever about (p,q) on that curve.

experimentx · Answer

a/p = -b/q since p,q lies in the curve it must satify
a/p - b/q = 1
=> a/p - b/q = 1 => 2a/p = 1

experimentx · Answer

Sorry for above, it does not lie in a/x + b/y = 1 they have different solutions http://www.wolframalpha.com/input/?i=solve+ax%2Bby%3D1%2C+a%2Fx-b%2Fy%3D1&dataset= http://www.wolframalpha.com/input/?i=solve+ax%2Bby%3D1%2C+a%2Fx%2Bb%2Fy%3D1&dataset=&equal=Submit

experimentx · Answer

that was just simple substitution that looked like a/x +b/y = 1

anonymous · Answer

Okay lets step back for a second and confirm that we're on the same ground.
- What does it means by normal to the curve?

experimentx · Answer

the product of slope must be -1

anonymous · Answer

Yes but what does it really means?
Can you describe it graphically?

experimentx · Answer

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