Question

point a lies on the circle and has an x-coordinate of -3. which value could be the y-coordinate for point a? \\(\sqrt{6}\\) \\(\sqrt{7}\\) 2.5 2.8

Explanation:

Step1: Determine the circle's equation

The circle is centered at \((0,0)\) (origin) and has a radius \(r = 4\) (since the distance from \((0,0)\) to \((-4,0)\) is \(4\)). The equation of a circle centered at \((h,k)\) with radius \(r\) is \((x - h)^2+(y - k)^2=r^2\). Here, \(h = 0\), \(k = 0\), \(r = 4\), so the equation is \(x^2 + y^2=16\).

Step2: Substitute \(x=-3\) into the equation

Substitute \(x=-3\) into \(x^2 + y^2=16\): \((-3)^2+y^2 = 16\), which simplifies to \(9 + y^2=16\).

Step3: Solve for \(y\)

Subtract 9 from both sides: \(y^2=16 - 9=7\). Then take the square root: \(y=\pm\sqrt{7}\). Since point \(A\) is in the second quadrant (from the graph), \(y\) is positive, so \(y = \sqrt{7}\).

Answer:

\(\sqrt{7}\)