A circle in the standard (x, y) coordinate plane is defined by the equation x^2 + y^2 = 10… — ACT

A circle in the standard $(x, y)$ coordinate plane is defined by the equation $x^2 + y^2 = 10$ . The line $y = 3x$ intersects the circle at a point $(x, y)$ in the first quadrant. What is the value of $x + y$ ?

Answer: 4

Answer

The correct answer is 4.

Substituting

y = 3x

into the equation

x^2 + y^2 = 10

yields

x^2 + (3x)^2 = 10

. Simplifying this expression gives

x^2 + 9x^2 = 10

, which combines to

10x^2 = 10

. Dividing both sides by 10 results in

x^2 = 1

. Since the point of intersection lies in the first quadrant, we take the positive square root, so

x = 1

. Substituting

x = 1

back into

y = 3x

gives

y = 3(1) = 3

. The sum of the coordinates is

x + y = 1 + 3 = 4

Step-by-Step Solution

Substitute the linear expression for

y

into the equation of the circle.

x^2 + (3x)^2 = 10

Substituting one equation into another allows us to solve for a single variable.

Simplify the equation and solve for

x

10x^2 = 10 \Rightarrow x^2 = 1 \Rightarrow x = 1

Combining like terms gives

10x^2 = 10

, which simplifies to

x^2 = 1

. Since the intersection point is in the first quadrant,

x

must be positive.

Substitute the value of

x

back into the linear equation to find

y

y = 3(1) = 3

Using the value of

x

in the linear equation determines the corresponding

y

-coordinate.

Calculate the sum of the coordinates

x + y

1 + 3 = 4

The question asks for the value of the sum of the coordinates of the intersection point.

Key Concept

Solving a system of linear and non-linear (circular) equations using substitution.

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