# Area of triangle bounded by a tangent to a curve

Find a curve such that the surface of the triangle bounded by the line going through the tangent point and perpendicular to the x-axis and the tangent line to the graph is equal to $$a^2$$.

I didn't understand the question at first glance, and I found related answers like this one - but this question also assumes the triangle is bounded by a perpendicular line to the $$x$$ axis.

This is what I got so far, but I don't think I got the DE entirely correct. I think I need some help with interpreting the question.

The tangent line to the curve at any point $$x$$ is given by:

$$y-xy^{\prime}=0$$

This line intersects the $$x$$ axis at some point $$x_{0}$$ s.t. $$y(x_0)=0$$. Then, for any $$x>x_{0}$$, a $$\perp$$ line of height $$y\left(x\right)$$ intersects the tangent line.

So the triangle is defined by:

$$A\left(x_{0},0\right)$$
$$B\left(x,0\right)$$
$$C\left(x,y-xy^{\prime}\right)$$

Thus, the area of $$\triangle ABC$$ is given by:

$$\frac{1}{2}\left(x-x_{0}\right)\left(y-xy^{\prime}\right) =a^{2}$$

• Are you sure the equation of tangent is correct? According to me, it should be $\frac{y-Y}{x-X}=g'(X)$ where $(X,Y)$ is the point of tangency and $g(x)$ is the desired curve. – Shubham Johri Nov 1 '20 at 20:49
• @ShubhamJohri But isn't the point of tangency should be a general point, not a particular one? How does $(X,Y)$ differ from the $(x,y)$ arguments in your equation? – gbi1977 Nov 1 '20 at 20:58

Let the point of tangency on the curve $$y=g(x)$$ be $$(X,g(X))$$.

Equation of tangent is $$\frac{y-g(X)}{x-X}=g'(X)$$.

$$x-$$intercept of tangent is $$X-g(X)/g'(X)$$.

The area of the triangle in question is $$\frac12\times|X-(X-g(X)/g'(X))|\times|g(X)|=a^2$$.

This gives the ODE$$|g'|=g^2/2a^2$$

Since we want to find any one solution, we let $$g'\ge0$$. Thus$$\int\frac{dg}{g^2}=-\frac1g=\frac x{2a^2}+c$$You can assume $$g'\le 0$$ and you would get $$1/g=x/2a^2+c_1$$ which is also admissible.

• I don't follow, why x−intercept of tangent is $X−\frac{g(X)}{g′(X)}$? – gbi1977 Nov 1 '20 at 21:07
• Just substitute $y=0$ in tangent eqn. @gbi1977 – Shubham Johri Nov 1 '20 at 21:08
• @gbi1977 Just remember that at $x=-2a^2c$ the function is not defined. So remove that from the domain. – Shubham Johri Nov 1 '20 at 21:11