Topic
In the coordinate plane Oxygenfor circle (OLD): (x + 2)2 + (y − 4)2 = 25 and points A(-1; 3).
a) Determine the relative position of the point A for the circle (OLD)
b) Straight line d change passes A intersect the circle at USA and WOMEN. Write the equation of the line d so that MN the shortest
Solution method – See details
Step 1: Determine the coordinates of the center I and radius CHEAP of (C)
Step 2: Compare the length IA and radius CHEAP to consider the relative position of A with (OLD)
Step 3: Apply the property of chords as far from the center as possible to find the GTLN of \(d(I,d)\)
Step 4: Write the PTQ of d with the elements found in step 3
Detailed explanation
a) (OLD) has a mind I(-2 ; 4) and radius CHEAP = 5
We have: \(\overrightarrow {IA} = (1; – 1) \Rightarrow IA = \sqrt 2 \)
Yes: \(IA = \sqrt 2 < R \Rightarrow \) Score A lies inside the circle (OLD)
b) According to the assumption, d cut (OLD) at 2 points USA, WOMEN satisfy MN shortest \( \Leftrightarrow \) distance from center I arrive d biggest
Call H is the projection of I above d. We have: \(IH \le IA\)
\( \Rightarrow \) IH achieves GTLN if and only if H coincides with A
\( \Rightarrow IA \bot d\) \( \Rightarrow d\) takes \(\overrightarrow {IA} = (1; – 1)\) as the normal vector, so we have PT: x – y + 4 = 0
