**Topic**

In the coordinate plane *Oxygen*for 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

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