# Tangent of a conic section

Let $$ax^2 + hxy + by^2 + gx + fy + c = 0$$ be the equation of a conic section

I want to find the equation of tangent and normal at some point on the curve say $$(x_0 , y_0)$$

I know there is some rule for changing the above equation to the equation of its tangent passing through the given point. For example: $$x$$ changes to $$\frac{x_0 + x}{2}$$ I don't know the other rules. I know it is something related to calculus. However I am still in class 11, so I only know the basics. Can you please give me the other rules.

## 1 Answer

Suppose x and y are continuously differentiable functions of a parameter t.For example ,imagine t is time and that (x(t),y(t)) is the position at time t of a bug crawling along the curve. Then $$0=\frac {d}{dt}(a x^2+ h x y+b y^2+g x+f y+c)=$$ $$2 a x x'+h x' y+h x y' +2 b y y'+g x'+f y',$$ $$\text{where }x'=dx/dt , \text{and } y'=dy/dt.$$ Re-grouping, we have $$x'(2a x+h y+g)=-y'(h x+2 b y+f).$$ One of the nice things about the notation $dx/dt$ is that in many (not all) cases we can treat the numerator and denominator as if they were numbers.So we expect $$dy/dx= \frac {dy/dt}{dx/dt}=y'/x'.$$ provided that $h x+2 b y+f\ne 0.$ From the previous equation ,and this,we can find the slope of the tangent line at $(x,y)$.