- #1

vkash

- 318

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Here i am asking all the things about parabola. I think something is wrong with proof where we prove condition of tangency.

It is done shown(in my book)

[itex]y^2=4ax[/itex] //equation of general parabola

let us assume that line [itex]y=mx+c[/itex] is passing through parabola.

points where it will cut parabola.

[itex](mx+c)^2=4ax[/itex]

solving this equation.

[itex](mx)^2+x(2mc-4a)+c^2=0[/itex]

line will touch parabola at one point if of the roots of this equation are equal.

It means it’s discriminant is zero.

[itex](2mc-4a)^2-4m^2c^2=0[/itex]

=> [itex]16a^2=16amc[/itex]

[itex]a=0 [/itex]; not possible because in that case it will not remain a parabola it became a line y=0.

this is the require condition for the line to cut the parabola at one point.

So let's take an example.

[itex]y=2[/itex]. this line cuts the parabola [itex]y^2-4x=0[/itex] at one point. as we can see on the graphs

but does it obey the equation proved previously.

a=0/2. NO. it is not obeying that equation.

WHY??

the condition is for line to cut the parabola at one point.y=2 is also a line that cuts parabola at one point but not obeying the condition.

It is done shown(in my book)

[itex]y^2=4ax[/itex] //equation of general parabola

let us assume that line [itex]y=mx+c[/itex] is passing through parabola.

points where it will cut parabola.

[itex](mx+c)^2=4ax[/itex]

solving this equation.

[itex](mx)^2+x(2mc-4a)+c^2=0[/itex]

line will touch parabola at one point if of the roots of this equation are equal.

It means it’s discriminant is zero.

[itex](2mc-4a)^2-4m^2c^2=0[/itex]

=> [itex]16a^2=16amc[/itex]

[itex]a=0 [/itex]; not possible because in that case it will not remain a parabola it became a line y=0.

**[itex]a=mc[/itex]**this is the require condition for the line to cut the parabola at one point.

So let's take an example.

[itex]y=2[/itex]. this line cuts the parabola [itex]y^2-4x=0[/itex] at one point. as we can see on the graphs

but does it obey the equation proved previously.

a=0/2. NO. it is not obeying that equation.

WHY??

the condition is for line to cut the parabola at one point.y=2 is also a line that cuts parabola at one point but not obeying the condition.

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