Questions: Find the x-coordinates of all points on the circle (x^2+y^2=53) with a y-coordinate of 7.

Solution Steps

To find the \( x \)-coordinates of all points on the circle \( x^2 + y^2 = 53 \) with a \( y \)-coordinate of 7, we can substitute \( y = 7 \) into the circle's equation and solve for \( x \). This will give us the possible \( x \)-coordinates.

1. Substitute \( y = 7 \) into the equation \( x^2 + y^2 = 53 \).
2. Solve the resulting equation for \( x \).

We start with the equation of the circle given by

\[ x^2 + y^2 = 53 \]

Substituting \( y = 7 \) into the equation, we have:

\[ x^2 + 7^2 = 53 \]

Calculating \( 7^2 \):

\[ x^2 + 49 = 53 \]

Now, we isolate \( x^2 \):

\[ x^2 = 53 - 49 \]

This simplifies to:

\[ x^2 = 4 \]

Taking the square root of both sides, we find:

\[ x = \sqrt{4} \quad \text{or} \quad x = -\sqrt{4} \]

Thus, we have:

\[ x = 2 \quad \text{or} \quad x = -2 \]

Final Answer