Question

question
in the figure above, which of the lettered points could be a point of intersection of the circle ( x^2 + y^2 = 25 ) and the line ( y = 3 )?
○ a
○ b
○ c
○ d

Explanation:

Step1: Find intersection points

To find the intersection of the circle \(x^{2}+y^{2} = 25\) and the line \(y=3\), substitute \(y = 3\) into the circle's equation.
\[x^{2}+3^{2}=25\]
\[x^{2}+9 = 25\]
\[x^{2}=25 - 9\]
\[x^{2}=16\]
\[x=\pm4\]
So the intersection points are \((4,3)\) and \((- 4,3)\). These points have a positive \(y\)-coordinate (\(y = 3\)) and \(x\)-coordinates \(\pm4\), so they lie in the first and second quadrants (since \(y>0\)).

Step2: Analyze the points

• Point \(A\): On the \(y\)-axis, \(x = 0\), \(y=5\) (since the circle has radius \(5\), center at origin). So \(A=(0,5)\), not on \(y = 3\).
• Point \(B\): In the second quadrant ( \(x<0,y>0\) ), which matches the intersection point \((-4,3)\) (since \(- 4<0,3>0\)).
• Point \(C\): On the \(x\)-axis, \(y = 0\), so not on \(y=3\).
• Point \(D\): In the first quadrant (\(x>0,y>0\)), but the \(x\)-coordinate of \(D\) looks less than \(4\) (since the point \((5,0)\) is on the \(x\)-axis, and \(D\) is between \((5,0)\) and \((0,5)\) but closer to \(x\)-axis? Wait, no, the intersection point in first quadrant is \((4,3)\). Let's check the position: the circle has radius \(5\), so at \(y = 3\), \(x=\pm4\). Point \(D\): if we look at the diagram, \(D\) is in the first quadrant, but is its \(x\)-coordinate \(4\)? Wait, point \(B\) is in the second quadrant, \(x\) negative, \(y\) positive. The intersection point \((-4,3)\) is in the second quadrant, so \(B\) is a candidate. Point \(D\): if \(x = 4\), \(y=3\), but in the diagram, \(D\) is closer to the \(x\)-axis? Wait, no, the line \(y = 3\) is a horizontal line. The \(y\)-coordinate of \(B\) is between \(0\) and \(5\), and \(x\) is negative. The intersection point \((-4,3)\) has \(x=-4,y = 3\), which is in the second quadrant, so \(B\) is the point.

Answer:

B