In the xy-plane, the circle with equation x^2 + y^2 = 13 intersects the line with equation… — SAT

In the $xy$ -plane, the circle with equation $x^2 + y^2 = 13$ intersects the line with equation $y - x = 1$ at two points, $(x_1, y_1)$ and $(x_2, y_2)$ . What is the value of $y_1 + y_2$ ?

A
-1
1Answer
C
3
D
-5

Answer

The sum of the

y

-coordinates of the intersection points is

1

The correct answer is

1

. Substituting

x = y - 1

from the linear equation into the circle equation yields

(y-1)^2 + y^2 = 13

. Expanding and simplifying gives the quadratic equation

2y^2 - 2y - 12 = 0

. Dividing by 2 yields

y^2 - y - 6 = 0

, which factors into

(y - 3)(y + 2) = 0

. Therefore, the

y

-coordinates of the intersection points are

3

and

-2

. The sum of these coordinates is

3 + (-2) = 1

Step-by-Step Solution

Rearrange the linear equation to solve for

x

in terms of

y

x = y - 1

Expressing

x

in terms of

y

allows us to substitute it into the circle equation to find the

y

-coordinates directly.

Substitute

x = y - 1

into the circle equation

x^2 + y^2 = 13

(y - 1)^2 + y^2 = 13

This substitution reduces the system of equations to a single quadratic equation in terms of

y

Expand and simplify the quadratic equation into standard form

ay^2 + by + c = 0

y^2 - 2y + 1 + y^2 = 13 \implies 2y^2 - 2y - 12 = 0

Expanding the squared term and combining like terms puts the equation in a form that can be solved.

Divide the equation by 2 and factor the simplified quadratic equation.

y^2 - y - 6 = 0 \implies (y - 3)(y + 2) = 0

Simplifying and factoring helps find the individual

y

-coordinates of the two intersection points.

Solve for

y

and calculate the sum of the coordinates.

y = 3

y = -2

, and

3 + (-2) = 1

Solving the factored equation gives the

y

-coordinates of the two intersection points, which are then added to find the final value.

Key Concept

Solving a nonlinear system of equations representing a circle and a line via substitution.

Alternative Method

Instead of solving for

x

first, we can substitute

y = x + 1

into the circle equation to find the

x

-coordinates:

x^2 + (x + 1)^2 = 13 \implies 2x^2 + 2x - 12 = 0 \implies x^2 + x - 6 = 0 \implies (x + 3)(x - 2) = 0

. This gives

x = -3

x = 2

. We then find the corresponding

y

-values using

y = x + 1

: for

x = -3

y = -2

; for

x = 2

y = 3

. The sum of the

y

-coordinates is

-2 + 3 = 1

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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