In the system of equations below, k is a constant. \begin{aligned} x^2 + y^2 + 8y &= 9 \\ … — SAT

In the system of equations below,

k

is a constant.

\begin{aligned} x^2 + y^2 + 8y &= 9 \\ x^2 + y &= k \end{aligned}

If the system has exactly 3 distinct real solutions

(x, y)

, what is the value of

k

Cevap: 1

Cevap

The value of

k

is 1.

Substituting

x^2 = k - y

into

x^2 + y^2 + 8y = 9

gives

y^2 + 7y + k - 9 = 0

. For the system to have exactly 3 distinct real solutions, the vertex of the parabola

y = -x^2 + k

must lie on the circle, which corresponds to the root

y = k

. Substituting

y = k

into the quadratic equation yields

k^2 + 8k - 9 = 0

, which gives

k = 1

k = -9

. For

k = 1

, the roots of the quadratic are

y = 1

and

y = -8

. The root

y = 1

yields 1 real solution,

(0, 1)

, and the root

y = -8

yields 2 real solutions,

(3, -8)

and

(-3, -8)

, for a total of 3 real solutions. For

k = -9

, the roots are

y = -9

and

y = 2

. The root

y = 2

does not yield any real solutions for

x

because

2 > -9

, so the system has only 1 real solution. Thus,

k = 1

Adım Adım Çözüm

Express

x^2

in terms of

y

and

k

from the second equation.

x^2 = k - y

(with constraint

y \le k

for real

x

)

To prepare for substitution into the first equation and establish the boundary condition for real solutions.

Substitute

x^2 = k - y

into the first equation.

y^2 + 7y + k - 9 = 0

To create a single quadratic equation in terms of

y

Analyze the conditions on the roots of the quadratic equation to get exactly 3 distinct real solutions.

One root must equal

k

and the other must be less than

k

A root

y = k

yields 1 real solution for

x

(

x = 0

), while a root

y < k

yields 2 real solutions (

x = \pm\sqrt{k-y}

Find the candidate values of

k

by setting

y = k

in the quadratic equation.

k^2 + 8k - 9 = 0 \implies k = 1

k = -9

To find the values of

k

where the parabola's vertex lies on the circle.

Verify which candidate value of

k

satisfies all conditions.

For

k = 1

, the roots are

y = 1

and

y = -8 < 1

(3 solutions). For

k = -9

, the roots are

y = -9

and

y = 2 > -9

(1 solution). Therefore,

k = 1

To ensure the second root is strictly less than

k

, guaranteeing exactly 3 solutions.

Anahtar Kavram

Analyzing the number of solutions in a nonlinear system of equations using algebraic substitution and boundary constraints.

Tahmini Süre:3m 0s

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