Solve Lesson 56 Page 89 SBT Math 10 – Kite>


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: xy + 4 = 0

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