What is the minimum length chord of a circle passing through a specific point?

[utkarshakash]

Homework Statement

px+qy=40 is a chord of minimum length of the circle $(x-10)^2 + (y-20)^2 = 729$. If the chord passes through (5,15), then $p^{2013}+q^{2013}$ is equal to

Homework Equations

The Attempt at a Solution

Let chord length be L

$\frac{L}{2} = 729- \dfrac{(10p+20q-40)^2}{p^2+q^2}$

Also
5p+15q-40=0

Now if I apply Lagrange's Multiplier Method using above two conditions I get some weird value of q which is a huge fraction.

 

[haruspex]

I don't think you need to be using a Lagrange Multiplier here.
Although it's not expressed clearly, it must be that the chord has the minimum length of all those passing through the given point. Isn't the implication clear from the geometry? Or was this given as an exercise in using Lagrange Multipliers?

 

[utkarshakash]

I don't think you need to be using a Lagrange Multiplier here.
Although it's not expressed clearly, it must be that the chord has the minimum length of all those passing through the given point. Isn't the implication clear from the geometry? Or was this given as an exercise in using Lagrange Multipliers?

From geometry I can see that minimum length will occur if the given line is parallel to Y-axis. So q=0 and p=8. The answer should then be 2^6039. But this does not match any of the options given.

 

[Mentallic]

From geometry I can see that minimum length will occur if the given line is parallel to Y-axis.

No, that's not correct. The minimum length would be a line parallel to the y-axis only if the point that the line passes through has a y value equal to that of the y value for the centre of the circle. That is, since the circle has centre at (10,20) then only lines that must pass through points on y=20 would be parallel to the y-axis to give a minimum length chord.

The minimum length chord passing through a point P is perpendicular to the line OP where O is the centre of the circle.