0
$\begingroup$

The equation of circle $C$ is $x^2 + (y − 8)^2 = 25$. The eye is located at $E = (0, −5)$. The maximal circular arc visible to the eye is $AB$, which is then being projected on to the one-dimensional "screen" as $A'B'$.

What are the co-ordinates of points $A$ and $B$?

I came this far: point $P$ on circle $C$ has the coordinates $x = 5 \cos\theta$, $y = 8 + 5 \sin \theta$. Now I should use this to find points $A$ and $B$, but I don't know how to proceed.

$\endgroup$
3
  • 1
    $\begingroup$ The points $A$ and $B$ should be the points of tangency to the lines drawn from the point $(0,-5)$. So find the equation of the tangent lines drawn from this point and you can get the tangency points as well. $\endgroup$
    – Anurag A
    May 18, 2020 at 17:16
  • $\begingroup$ @AnuragA I'm still stuck and not sure how to solve it, would you be able to write it out? $\endgroup$
    – John Hm
    May 18, 2020 at 17:26
  • $\begingroup$ @AnuragA It’s much less work in general to compute the polar of the point and intersect that single line with the circle. The symmetry of the problem requires only one tangent line, if you’re going that route. $\endgroup$
    – amd
    May 18, 2020 at 18:14

5 Answers 5

0
$\begingroup$

I will use your parametric representation. Observe that the center of this circle is at $C(0,8)$. The distance $|CE|=8+5=13$. The $\triangle CAE$ is a right angled triangle with $|CA|=5$ (radius) and $|CE|=13$, so $EA=12$ (Pythagoras) (same will be true for $EB$).

Now $$|EA|=12 \implies \sqrt{(0-5 \cos \theta)^2+(-5-8-5\sin \theta)^2}=12.$$ This gives us $$194+130 \sin \theta=144. \implies \sin \theta=\frac{-5}{13}.$$ Thus we also get $\cos \theta=\pm \frac{12}{13}$.

So $A,B=(5 \cos \theta, 8+5\sin \theta)=\left(\pm \frac{60}{13}, \frac{79}{13}\right)$

$\endgroup$
0
$\begingroup$

This is easier to solve, I think, using the original Cartesian equation. The key observation for any solution is that the endpoints of the visible arc are also the endpoints of the chord of contact of the two tangents to $C$ through $E$.

The nice layout of $E$ and $C$ with respect to the coordinate axes makes for a relatively simple solution using similar triangles, but I’ll continue with a general method. The chord of contact is the polar of $E$ with respect to $C$. If the equation of a circle is $(x-h)^2+(y-k)^2=r^2$, then the polar of the point $(x_0,y_0)$ has the equation $(x-h)(x_0-h)+(y-k)(y_0-k)=r^2$. Plugging in the values from this problem, the polar of $E$ is $(y-8)(-5-8)=25$, or $y=\frac{79}{13}$. Intersect this line with the circle to find $A$ and $B$, which for this problem is a simple matter of substituting into the circle’s equation and solving for $x$.

$\endgroup$
0
$\begingroup$

Note that $O(0,8)$ is the center, and EAO and EBO are right triangles with AB $\perp$ EO, giving

$$EA=EB= \sqrt{EO^2-r^2} =\sqrt{(8+5)^2 -5^2}=12$$

and $AB =2 \frac{ EA \cdot OA }{EO}= \frac{2\cdot 12\cdot 5}{13}=\frac{120}{13}$. So, let $A(\frac{60}{13},b)$, $B(-\frac{60}{13},b)$ and substitute them into $x^2+(y-8)^2=25$ to obtain $b=\frac{79}{13}$.

Thus, the endpoints are $(\pm \frac{60}{13},\frac{79}{13})$.

$\endgroup$
0
$\begingroup$

Another way (non calculus) using only the theory of quadratics is to simultaneously solve the equation of the circle and the lines of tangency.

The circle has equation $x^2 + (y-8)^2 = 25$, while the lines of tangency have the form $y =mx - 5$, where $m$ is the slope (to be found).

We get a single variable quadratic in $x$, whose discriminant is $100m^2 - 576$. For tangency, the discriminant must vanish (so you get a single repeated root at point of contact). Then you get $m = \pm \frac{12}5$ from which you can get the points of tangency as $(\pm \frac{60}{13}, \frac{79}{13})$.

$\endgroup$
0
$\begingroup$

Note - the solution relies on using AM$\geqslant$ GM inequality without proving it , check this link out if you are not familiar with this inequality Proofs of AM-GM inequality

Let's first initialise parametric coordinates for the circle $x^2 + (y-8)^2 = 25$ as $P= (5 cos \theta , 8 + 5 \sin \theta )$

Let the given point be $Q(0,-5)$

circle and the point plotted

Then find slope of line segment PQ M($ \theta $)=$ \frac {13 + 5 \sin ( \theta)}{5 \cos ( \theta )}$ Which simplifies to$ \frac {13}{5} \sec( \theta ) + tan(\theta)$

We now find range of M Let $\sec( \theta) + \tan( \theta) = t$ Then $\sec(\theta) - tan(\theta) = \frac{1}{t} $(from trigonometric identites)

t lies on ( -$ \infty , + \infty$ )

Now , $\sec(\theta) = (t + \frac{1}{t} )/2$ And $tan(\theta) = (t - \frac{1}{t} )/2$

Putting it in the slope function M and simplifying M = $\frac{18t}{10} + \frac{8}{10t}$ On positive range of t , applying AM $\geqslant$ GM $$\frac{18t}{10} + \frac{8}{10t} \geq \sqrt{\frac{18t}{10} *\frac{8}{10t}}$$ $$M \geq \frac{12}{5} $$ Similarly , $M \leq -\frac{12}{5} $( to prove this assume t is -ve , then $-t$ will be positive , apply $AM \geq GM$ for this )

equality holds good for$ \frac{18t}{10} = \frac{8}{10t}) $ (in both the cases (namely t>0 and t<0 )

Hence t = +12/5 or -12/5

Using these to get $ \sec(\theta)$ and hence find the end points by putting the values of \theta in the circle's parametric coordinates .

The arc required is the lower side part of the circle enclosed by the 2 lines

$\endgroup$
1
  • $\begingroup$ you are seeing how to fully convert to complete latex, if you complete the whole answer into it, you'll receive upvotes for sure, I will do it $\endgroup$
    – janmarqz
    May 18, 2020 at 19:32

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .