From the point A (0, 3) on the circle x² + 4x + (y - 3)² = 0 a chord AB is drawn & extended to a point M such that AM = 2 AB. The equation of the locus of M is :

The given equation x² + 4x + ( y-3 )² = 0 can be written as

( x+2 )² + ( y-3 )² = 2².

It represents the circle with C ≡ ( -2, 3 ) and r = 2.

It touches Y-axis at the point A ≡ ( 0, 3 ).

Let B( x₁, y₁ ) be a point on this circle so that

(x₁ + 2)² + (y₁ - 3)² = 2², i.e.,

(x₁)² + (y₁)² + 4(x₁) - 6(y₁) + 9 = 0 ......... (1)

If M(h,k) is a point such that AM = 2(AB),

i.e., AB = BM, then B is the mid-point of AM.

Then, B ≡ ( (0+h)/2, (3+k)/2 ) ≡ ( x₁, y₁ )

i.e., (h)/2 = x₁ ... and ... (3+k)/2 = y₁

Hence, from (1),

(h/2)² + [(3+k)/2]² + 4(h/2) - 6[(3+k)/2] + 9 = 0

(h²/4) + [(3+k)²/4] + 2h - 3(3+k) + 9 = 0

h² + (3+k)² + 8h - 12(3+k) + 36 = 0

h² + 9 + 6k + k² + 8h - 36 - 12k + 36 = 0

h² + k² + 8h - 6k + 9 = 0.

Replacing the current co-ordinates (h,k) by the general co-ordinates (x,y),

the required equation of the locus of M is

x² + y² + 8x - 6y + 9 = 0

that is ( x + 4 )² + ( y - 3 )² = 4².
The text passage describes a geometric problem involving a circle, a chord, and the locus of a point. It provides a detailed solution to find the equation of the locus of point M.
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Given a text passage, determine:
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    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

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TYPE: <one of the following values>
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From the point A (0, 3) on the circle x² + 4x + (y - 3)² = 0 a chord AB is drawn & extended to a point M such that AM = 2 AB. The equation of the locus of M is :

The given equation x² + 4x + ( y-3 )² = 0 can be written as

( x+2 )² + ( y-3 )² = 2².

It represents the circle with C ≡ ( -2, 3 ) and r = 2.

It touches Y-axis at the point A ≡ ( 0, 3 ).

Let B( x₁, y₁ ) be a point on this circle so that

(x₁ + 2)² + (y₁ - 3)² = 2², i.e.,

(x₁)² + (y₁)² + 4(x₁) - 6(y₁) + 9 = 0 ......... (1)

If M(h,k) is a point such that AM = 2(AB),

i.e., AB = BM, then B is the mid-point of AM.

Then, B ≡ ( (0+h)/2, (3+k)/2 ) ≡ ( x₁, y₁ )

i.e., (h)/2 = x₁ ... and ... (3+k)/2 = y₁

Hence, from (1),

(h/2)² + [(3+k)/2]² + 4(h/2) - 6[(3+k)/2] + 9 = 0

(h²/4) + [(3+k)²/4] + 2h - 3(3+k) + 9 = 0

h² + (3+k)² + 8h - 12(3+k) + 36 = 0

h² + 9 + 6k + k² + 8h - 36 - 12k + 36 = 0

h² + k² + 8h - 6k + 9 = 0.

Replacing the current co-ordinates (h,k) by the general co-ordinates (x,y),

the required equation of the locus of M is

x² + y² + 8x - 6y + 9 = 0

that is ( x + 4 )² + ( y - 3 )² = 4².
user
The text passage describes a geometric problem involving a circle, a chord, and the locus of a point. It provides a detailed solution to find the equation of the locus of point M.

TYPE: SINGLE_QUESTION_WITH_ANSWER