# Tangent line

1. Aug 2, 2013

### dirk_mec1

1. The problem statement, all variables and given/known data
A tangent line at point A with coordinate (a,f(a)) of function f(x) intersects f(x) at point B coordinate (b,f(b)) . A vertical line is drawn from point p (a<p<b) and intersects f(x) at C. From C a perpendicular line to the tangent line is drawn which intersect the tangent line at point D with coordinate (q,f(q))

The tangent line can be described by $y_1(x) =a_1 x+ b_1$

|CD| can be described by $y_2(x) =a_2 x+ b_2$

Express q in terms of $a_1,b_1,a_2,b_2$

[Broken]

2. Relevant equations
-

3. The attempt at a solution
I can find $a_1,b_1,a_2,b_2$ in terms of a, b, f(a) and f(b).

I can express q in terms of the slope of the tangent:

$$q = p + H \sin(\alpha) \cos(\alpha)$$

with
$$\alpha = arctan\left( \frac{f(b)-f(a)}{b-a} \right)$$

$$H = f(p) -y_1(p)$$
and now what?

Last edited by a moderator: May 6, 2017
2. Aug 2, 2013

### Mandelbroth

This is a fun Euclidean geometry problem.

$a_1q+b_1=a_2q+b_2$. Solve for $q$.

Last edited by a moderator: May 6, 2017
3. Aug 2, 2013

### verty

I wonder if there is a mistake in this question. Expressing q in terms of $a_1$, $a_2$, $a_3$ and $a_4$ means we don't need to know what those numbers are.

4. Aug 4, 2013

### dirk_mec1

I can seriously hit myself against the wall. Thanks man.

5. Aug 4, 2013

### Mandelbroth

No problem. They threw a lot of unnecessary information out there. It took me a minute too. :tongue: