A circle and a line are graphed in the xy-plane. The circle is defined by the equation x^2… — SAT

A circle and a line are graphed in the $xy$ -plane. The circle is defined by the equation $x^2 + y^2 = 25$ , and the line is defined by the equation $y = 3$ . If the line intersects the circle at the point $(x, 3)$ , where $x > 0$ , what is the value of $x$ ?

Answer: 4

Answer

The correct answer is 4.

Substituting

y = 3

into the circle equation

x^2 + y^2 = 25

yields

x^2 + 3^2 = 25

. Simplifying this gives

x^2 + 9 = 25

, which simplifies to

x^2 = 16

. Taking the square root of both sides gives

x = 4

x = -4

. Since it is given that

x > 0

, the value of

x

must be

4

Step-by-Step Solution

Substitute the value of

y = 3

into the circle's equation.

x^2 + 3^2 = 25

Since the line is

y = 3

, any point of intersection must satisfy this y-coordinate. Substituting it into the circle's equation allows us to solve for the x-coordinate.

Simplify the equation and isolate

x^2

x^2 = 16

Squaring

3

gives

9

, and subtracting

9

from both sides of the equation

x^2 + 9 = 25

isolates

x^2

Solve for

x

and apply the constraint

x > 0

x = 4

Taking the square root of both sides of

x^2 = 16

gives

x = 4

x = -4

. Since the problem states

x > 0

, the only valid solution is

4

Key Concept

Solving systems of nonlinear equations using substitution

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