A circle in the xy-plane is defined by the equation (x - 2)^2 + (y + 1)^2 = 10. The line y… — SAT

A circle in the $xy$ -plane is defined by the equation $(x - 2)^2 + (y + 1)^2 = 10$ . The line $y = 3x + k$ , where $k$ is a constant, is tangent to the circle. If $k < 0$ , what is the value of $k$ ?

A
-3
B
-5
-17Answer
D
3

Answer

The correct value of

k

-17

Substituting the line equation

y = 3x + k

into the circle equation yields

(x - 2)^2 + (3x + k + 1)^2 = 10

. Expanding and writing this in standard form gives

10x^2 + (6k + 2)x + (k^2 + 2k - 5) = 0

. For the line to be tangent to the circle, the quadratic equation must have exactly one real solution, meaning its discriminant must be

0

. Setting the discriminant

\Delta = (6k+2)^2 - 4(10)(k^2 + 2k - 5)

0

and simplifying gives

-4k^2 - 56k + 204 = 0

. Dividing by

-4

yields

k^2 + 14k - 51 = 0

, which factors as

(k+17)(k-3)=0

. Since

k < 0

, the value of

k

must be

-17

Step-by-Step Solution

Substitute the linear equation

y = 3x + k

into the circle equation.

(x - 2)^2 + (3x + k + 1)^2 = 10

To find the coordinates where the line and the circle intersect.

Expand both squared terms and simplify the equation to standard quadratic form

Ax^2 + Bx + C = 0

10x^2 + (6k + 2)x + (k^2 + 2k - 5) = 0

To write the system as a single quadratic equation in terms of

x

Set the discriminant of the quadratic equation to zero.

(6k + 2)^2 - 4(10)(k^2 + 2k - 5) = 0

which simplifies to

-4k^2 - 56k + 204 = 0

Since the line is tangent to the circle, there must be exactly one intersection point, which means the quadratic equation must have exactly one real solution.

Divide the simplified equation by

-4

and solve for

k

k^2 + 14k - 51 = 0 \implies (k + 17)(k - 3) = 0 \implies k = -17 \text{ or } k = 3

To find the values of

k

that make the line tangent to the circle.

Apply the given constraint

k < 0

k = -17

The problem specifies that

k

must be a negative value.

Key Concept

Solving nonlinear systems of equations involving circles and lines by substitution and using the discriminant to determine tangency.

Estimated Time:2m 30s

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