To find the coordinates of the point [tex]\( P \)[/tex] where the tangents to the circle [tex]\( x^2 + y^2 = 25 \)[/tex] at points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] meet, we follow these steps:
1. Identify the circle's properties:
The given circle equation is [tex]\( x^2 + y^2 = 25 \)[/tex]. This represents a circle centered at the origin [tex]\((0, 0)\)[/tex] with radius [tex]\( 5 \)[/tex].
2. Find the equation of the tangent at point [tex]\( A \)[/tex]:
The coordinates of point [tex]\( A \)[/tex] are [tex]\((0, 5)\)[/tex]. The general equation of the tangent to a circle [tex]\( x^2 + y^2 = r^2 \)[/tex] at the point [tex]\((x_1, y_1)\)[/tex] is:
[tex]\[
x \cdot x_1 + y \cdot y_1 = r^2
\][/tex]
Substituting [tex]\( x_1 = 0 \)[/tex], [tex]\( y_1 = 5 \)[/tex], and [tex]\( r^2 = 25 \)[/tex]:
[tex]\[
x \cdot 0 + y \cdot 5 = 25 \implies y = 5
\][/tex]
Therefore, the tangent line at [tex]\( A \)[/tex] is [tex]\( y = 5 \)[/tex].
3. Find the equation of the tangent at point [tex]\( B \)[/tex]:
The coordinates of point [tex]\( B \)[/tex] are [tex]\((3, -4)\)[/tex]. Using the same tangent formula as before:
[tex]\[
x \cdot 3 + y \cdot (-4) = 25 \implies 3x - 4y = 25
\][/tex]
Therefore, the tangent line at [tex]\( B \)[/tex] is [tex]\( 3x - 4y = 25 \)[/tex].
4. Solve the system of equations to find the intersection point [tex]\( P \)[/tex]:
We now have the equations of the tangents:
[tex]\[
\begin{cases}
y = 5 \\
3x - 4y = 25
\end{cases}
\][/tex]
Substitute [tex]\( y = 5 \)[/tex] into the second equation:
[tex]\[
3x - 4(5) = 25 \implies 3x - 20 = 25 \implies 3x = 45 \implies x = 15
\][/tex]
Therefore, the coordinates of [tex]\( P \)[/tex] are:
[tex]\[
(x, y) = (15, 5)
\][/tex]
So, the intersection point [tex]\( P \)[/tex] where the tangents at points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] meet is [tex]\((15, 5)\)[/tex].
