Discriminant of Quadratic Equation

Question

If I have a line $y=mx+c$ tangent to parabola $y^2=4ax$ and I want to find the value of c in the tangent line equation

As per my teacher, he first made the substitution $x=\frac{y^2}{4a}$ in the line equation and had the whole equation rearranged to obtain :

$$y^2-(\frac{4a}{m})y + (\frac{4a}{m})c=0$$

Then his next step was to equate the discriminant of this equation to 0 to obtain c

from which I got $c=\frac{1}{m}$

and the equation of the tangent line turned out to be $y=mx+\frac{1}{m}$

My question is why did he get me to equate the discriminant to 0? What is the geometric interpretation of the discriminant of the quadratic equation? I know that when the discriminant is zero the quadratic equation has two identical real roots. Why is it imperative that the equation obtained have identical real roots?

Answer 1

Suppose a parabola of the type P : $y^2=4ax$. Let an arbitrary line L be $y=mx+c$.

Also,

$$y^2=m^2x^2+c^2+2mcx$$

$$4ax=m^2x^2+c^2+2mcx$$

$$m^2x^2+c^2+(2mc-4a)x=0\tag1$$

$$\Delta=(2mc-4a)^2-4m^2c^2=16a^2-16amc$$

Now for (1), you can interpret three cases, namely $(a)\Delta=0$ $(b)\Delta<0$ and $(c)\Delta>0$

Now to interpret geometrically based on the three cases of the discriminant $\Delta$.

case (a) $\Delta=0$, This implies that the quadratic equation posses a single root (or common root). This geometrically implies the two curves that were 'solved' by substituting into each other meet only once at an unique point. This is the condition you are to look for when in need for tangent as a tangent line meets a curve at one unique point only.

case (b) $\Delta>0$, This implies that the quadratic equation posses two real roots. This geometrically implies the two curves meet at two unique points as both curves solve to give two unique coordinates (for $x$ in this case). The line passes through the parabola at two different points (only).

case (c) $\Delta<0$, This implies that the quadratic equation posses complex roots. This geometrically implies the two curves meet at an "imaginary point", meaning the two curves lie on real plane and do not virtually posses an unique point of intersection.

Extras(1) - To complete form for the tangent equation you have derived is $$\boxed{y=mx+\frac{a}{m}}$$

Extras(2) - The condition for a line $y=mx+c$ to be tangent to a parabola $y^2=4ax$ is; $$\boxed{c=\frac{a}{m}}$$

Extras(3) - The point of contact is; $$\boxed{\left(\frac{a}{m^2},\frac{2a}{m}\right)}$$

Note - A polynomial equation of degree $n$ can atmost have $n$ roots. A quadratic equation can have two real roots, 1 real root and no real roots or two complex roots, as complex roots occur in conjugate pairs.