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Zorluk: ZorNonlinear Systems of Equations

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

  1. A
    -3
  2. B
    -5
  3. -17Cevap
  4. D
    3

Cevap

The correct value of kk is −17-17.
Substituting the line equation y=3x+ky = 3x + k into the circle equation yields (x−2)2+(3x+k+1)2=10(x - 2)^2 + (3x + k + 1)^2 = 10. Expanding and writing this in standard form gives 10x2+(6k+2)x+(k2+2k−5)=010x^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 00. Setting the discriminant Δ=(6k+2)2−4(10)(k2+2k−5)\Delta = (6k+2)^2 - 4(10)(k^2 + 2k - 5) to 00 and simplifying gives −4k2−56k+204=0-4k^2 - 56k + 204 = 0. Dividing by −4-4 yields k2+14k−51=0k^2 + 14k - 51 = 0, which factors as (k+17)(k−3)=0(k+17)(k-3)=0. Since k<0k < 0, the value of kk must be −17-17.

Adım Adım Çözüm

1
Substitute the linear equation y=3x+ky = 3x + k into the circle equation.
(x−2)2+(3x+k+1)2=10(x - 2)^2 + (3x + k + 1)^2 = 10
To find the coordinates where the line and the circle intersect.
2
Expand both squared terms and simplify the equation to standard quadratic form Ax2+Bx+C=0Ax^2 + Bx + C = 0.
10x2+(6k+2)x+(k2+2k−5)=010x^2 + (6k + 2)x + (k^2 + 2k - 5) = 0
To write the system as a single quadratic equation in terms of xx.
3
Set the discriminant of the quadratic equation to zero.
(6k+2)2−4(10)(k2+2k−5)=0(6k + 2)^2 - 4(10)(k^2 + 2k - 5) = 0 which simplifies to −4k2−56k+204=0-4k^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.
4
Divide the simplified equation by −4-4 and solve for kk.
k2+14k−51=0  ⟹  (k+17)(k−3)=0  ⟹  k=−17 or k=3k^2 + 14k - 51 = 0 \implies (k + 17)(k - 3) = 0 \implies k = -17 \text{ or } k = 3
To find the values of kk that make the line tangent to the circle.
5
Apply the given constraint k<0k < 0.
k=−17k = -17
The problem specifies that kk must be a negative value.

Anahtar Kavram

Solving nonlinear systems of equations involving circles and lines by substitution and using the discriminant to determine tangency.
Tahmini Süre:2m 30s
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Konular

Nonlinear Systems of Equations

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