Answered

Does the point $(-1, \sqrt{15})$ lie on the circle centered at the origin and containing the point $(0,4)$? Explain.


ANo, because the equation for the circle is $x^2+y^2=4$ and $(-1)^2+(\sqrt{15})^2

eq 4$.

BYes, because the equation for the circle is $x^2+y^2=16$ and $(-1)^2+(\sqrt{15})^2=16$.
CYes, because the equation for the circle is $x^2+y^2=4$ and $(-1)^2+(\sqrt{15})^2=4$.
DNo, because the equation for the circle is $x^2+y^2=16$ and $(-1)^2+(\sqrt{15})^2

eq 16$.

Verified Answer :

  • The circle's equation is determined to be \(x^2 + y^2 = 16\) based on its center at the origin and the point \((0,4)\).
  • We test the point \((-1, \sqrt{15})\) by substituting it into the equation \((-1)^2 + (\sqrt{15})^2 = 1 + 15 = 16\).
  • Since the substitution results in 16, which is equal to the radius squared, the point lies on the circle.
  • The correct option is b. \(\boxed{b}\)

Explanation

  1. Determine the equation of the circle.

The circle is centered at the origin \((0,0)\). The general equation for such a circle is \(x^2 + y^2 = r^2\), where \(r\) is the radius. We are given that the circle contains the point \((0,4)\). To find the radius squared (\(r^2\)), we substitute the coordinates of this point into the general equation:
\(0^2 + 4^2 = r^2\)
\(0 + 16 = r^2\)
\(r^2 = 16\)
Thus, the equation of the circle is \(x^2 + y^2 = 16\).

  1. Check if the point lies on the circle.

Now we check if the point \((-1, \sqrt{15})\) satisfies the circle's equation \(x^2 + y^2 = 16\). Substituting \(x = -1\) and \(y = \sqrt{15}\) into the left side of the equation gives:
\((-1)^2 + (\sqrt{15})^2\)
Calculating the squares:
\(1 + 15 = 16\)
Since the result is equal to 16, the coordinates of the point \((-1, \sqrt{15})\) satisfy the equation of the circle. Therefore, the point lies on the circle. Comparing this with the given options, this matches option b.

Examples

A city planner is designing a circular fountain centered at the city square, represented by the origin \((0,0)\). They know the edge of the fountain must pass through the point \((0,4)\) to align with a specific walkway. By calculating the circle's equation, they can determine if a proposed decorative statue located at \((-1, \sqrt{15})\) will be positioned directly on the edge of the fountain. This helps in precise urban design, ensuring that structures and landscapes are perfectly aligned according to the geometric plans.

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