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Step-by-Step Solution
Step 1: Represent the point in coordinates
Let the point P be (h, k) . We are given two fixed points: A(5,0) and B(-5,0) .
Step 2: Express the given distance condition
The distance from P to A is thrice the distance from P to B :
PA = 3 \times PB.
Squaring both sides gives:
PA^2 = 9 \, PB^2.
Step 3: Write distance formulas and equate
The distance PA is given by
PA^2 = (h - 5)^2 + k^2,
and the distance PB is
PB^2 = (h + 5)^2 + k^2.
Hence, from PA^2 = 9\,PB^2 , we get:
(h - 5)^2 + k^2 = 9\bigl((h + 5)^2 + k^2\bigr).
Step 4: Expand and simplify
Expand each term:
(h - 5)^2 = h^2 - 10h + 25
(h + 5)^2 = h^2 + 10h + 25
So the equation becomes:
h^2 - 10h + 25 + k^2
=
9 \bigl(h^2 + 10h + 25 + k^2\bigr).
Distribute the factor of 9 on the right side:
h^2 - 10h + 25 + k^2
=
9h^2 + 90h + 225 + 9k^2.
Bring all terms to one side:
0
=
9h^2 - h^2 + 90h + 10h + 225 - 25 + 9k^2 - k^2
\quad\Longrightarrow\quad
8h^2 + 100h + 200 + 8k^2
=
0.
Step 5: Put the equation into a standard form
Divide through by 8 :
h^2 + k^2 + \frac{100}{8}h + 25 = 0
\quad\Longrightarrow\quad
h^2 + k^2 + \frac{25}{2}\,h + 25
=
0.
Rewriting with standard geometry notation (x,y) instead of (h,k) , we get:
x^2 + y^2 + \frac{25}{2}x + 25 = 0.
Step 6: Identify the center and the radius
The general equation of a circle is
x^2 + y^2 + Dx + E = 0,
whose center is \bigl(-\frac{D}{2},\,0\bigr) when there is no y -linear term. Here, D = \frac{25}{2} and E = 25 . Thus the center is
\left(-\frac{25/2}{2},\,0\right)
=
\left(-\frac{25}{4},\,0\right).
The radius r satisfies
r^2
=
\left(-\frac{25}{4}\right)^2 - 25
=
\frac{625}{16} - 25
=
\frac{625}{16} - \frac{400}{16}
=
\frac{225}{16}.
Hence,
r = \sqrt{\frac{225}{16}}
=
\frac{15}{4}.
Step 7: Calculate 4r^2
Since r^2 = \frac{225}{16} :
4r^2
=
4 \times \frac{225}{16}
=
\frac{900}{16}
=
\frac{225}{4}
=
56.25.
If the problem requires an integer approximation, we take 56 (rounded).
Final Answer
The value of 4r^2 is 56.25 , which is often approximated as 56 if rounded to the nearest integer.
