Comparing equations to figure out coefficients of tangent line of conics

Question

If a focal chord with positive slope $m$ of the parabola $y^2 =16x$ touches the circle $x^2 +y^2-12x+34=0$ then the value of m is...?

I again use the result and I get equation of tangent of circle as (using result from page-91):

$$ Yy + X(x-6) + 68= 0$$

Or,

$$ Y + X \frac{x-6}{y} + \frac{68}{y}=0$$

Now, we can find equation of focal chord satisfying condition as $y=mx-4m$ or:

$$ y-mx+4m=0$$

Comparing,

$$ -m = \frac{x-6}{y}$$

And $$4m = \frac{68}{y}$$ But this leads to wrong answers!

I get $$x=23$$

Which is a point not even on the circle!

What is the mistake in my logic of applying this?

score 1 · Accepted Answer · answered Apr 10 '21 at 18:57

1

Equation of circle is $x^2 +y^2-12x+34=0$ or $(x-6)^2 + y^2 = 2$

The focal chord of the given parabola is $y = m(x-4)$ as you wrote.

The link in your question is not working for me but the easiest is to drop a perpendicular from center of the circle $(6, 0)$ to focal chord $y = m(x-4)$ and equate it to radius of the circle, $\sqrt2$ as the focal chord is tangent to the circle.

That gives $\frac{|0 - 6m + 4m|}{\sqrt{1+m^2}} = \sqrt2 \implies m = \pm 1$. As question says positive slope, we have $m = 1$.

Even by your method where it seems you are first finding equation of tangent to the circle and equating it to the focal chord,

Given the equation of the circle, taking derivative with respect to $x$, $2x + 2y y' - 12 = 0 \implies y' = \frac{6-x}{y}$

So tangent to circle at a given point $(x_0, y_0)$ is

$y - y_0 = \frac{6-x_0}{y_0} (x-x_0)$

But we also have $y = m (x - 4)$.

Solving them gives $x_0 = 5, y_0 = \pm1, m = \pm1$ and we consider $m = 1$.

answered Apr 10 '21 at 18:57

Math Lover

51,819

Umm yes, this is indeed the direct method but this doesn't explain me what the mistake in my work was >.< – tryst with freedom Apr 10 '21 at 18:58
I showed both method. The link you shared is not working for me. I can check otherwise. – Math Lover Apr 10 '21 at 18:59
Just a moment, I'll double check – tryst with freedom Apr 10 '21 at 18:59
I fixed the link, if you could check now it'd be greatly appreciated @Math Lover (p.s: Thanks for pointing out that link was broken!) – tryst with freedom Apr 10 '21 at 19:01
Equation of the tangent to circle should be $Y - y = \frac{6-x}{y} (X-x)$ as in my answer. I am not sure how you got the equation as $Y + X \frac{x-6}{y} + \frac{68}{y}=0$ – Math Lover Apr 10 '21 at 19:03
ok will check now. – Math Lover Apr 10 '21 at 19:03
1

Just when you wrote the equation of tangent line again I saw what my mistake was – tryst with freedom Apr 10 '21 at 19:04
ok great, saves me some time :) – Math Lover Apr 10 '21 at 19:05

score 1 · Answer 2 · answered Apr 10 '21 at 18:57

1

The equation of circle is $$(x-6)^2+y^2=2$$ so your $T=0$ equation is incorrect.

After finding the equation of focal chord as $$mx-y-4m=0$$

you can also set its distance from center of circle equal to the radius. $$\left( \frac{6m-0-4m}{\sqrt{m^2+1}} \right)^2=2$$

answered Apr 10 '21 at 18:57

cosmo5

10,629

Thank you, but same comment as I had said on math Lovers post – tryst with freedom Apr 10 '21 at 18:58
@Buraian I did explain that your first line is incorrect. – cosmo5 Apr 10 '21 at 18:59

score 1 · Answer 3 · edited Apr 10 '21 at 20:45

1

$$ Yy + X(x-6) + 68= 0$$

As pointed out by Math Lover, the above is incorrect. This is due to me not homogenizing the equation of circle properly, the correct equation would be:

$$x^2 + y^2-12xz+34z^2=0$$

Now taking derivatives:

$$ X[2x-12z] + Y[2y] + (-12x+68z)=0$$

$z=1$

$$ X[2x-12] + Y[2y] + (-12x+68) = 0$$

Which is the correct equation.

edited Apr 10 '21 at 20:45

Math Lover

51,819

answered Apr 10 '21 at 19:06

tryst with freedom

11,538

1

Thank you @MathLover ^^ Fixed it now – tryst with freedom Apr 10 '21 at 20:40
Oh, you are too kind. Thank you for fixing my error again. That was clumsy on my part. – tryst with freedom Apr 10 '21 at 20:47
that's alright, no problem – Math Lover Apr 10 '21 at 20:47

Comparing equations to figure out coefficients of tangent line of conics

3 Answers3