Examkin

Question

Difficulty: Very hardNonlinear Systems of Equations

For a constant kk, the circle (x−5)2+(y−5)2=18(x - 5)^2 + (y - 5)^2 = 18 and the line y=kxy = kx are graphed in the xyxy-plane. For how many integer values of kk will the circle and the line intersect at exactly two points?

  1. A
    5
  2. 6Answer
  3. C
    7
  4. D
    8

Answer

6
The correct answer is 6. By substituting the line equation y=kxy = kx into the circle equation, we obtain the quadratic equation (k2+1)x2−10(k+1)x+32=0(k^2 + 1)x^2 - 10(k + 1)x + 32 = 0. For the circle and line to intersect at exactly two points, this equation must have two distinct real roots, meaning its discriminant must be positive. This leads to the inequality 7k2−50k+7<07k^2 - 50k + 7 < 0, which factors as (7k−1)(k−7)<0(7k - 1)(k - 7) < 0. The solution is the interval 17<k<7\frac{1}{7} < k < 7. The integer values of kk in this range are 1,2,3,4,5,1, 2, 3, 4, 5, and 66, giving a total of 6 integers.

Step-by-Step Solution

1
Substitute the line equation y=kxy = kx into the circle equation (x−5)2+(y−5)2=18(x - 5)^2 + (y - 5)^2 = 18.
(x−5)2+(kx−5)2=18(x - 5)^2 + (kx - 5)^2 = 18
Substitution reduces the system of equations to a single quadratic equation in xx, representing the xx-coordinates of the intersection points.
2
Expand the terms and write the equation in standard quadratic form Ax2+Bx+C=0Ax^2 + Bx + C = 0.
(k2+1)x2−10(k+1)x+32=0(k^2 + 1)x^2 - 10(k + 1)x + 32 = 0
Putting the equation in standard form is necessary to analyze its discriminant.
3
Set the discriminant Δ=b2−4ac\Delta = b^2 - 4ac strictly greater than zero to ensure exactly two distinct real solutions.
[−10(k+1)]2−4(k2+1)(32)>0⇒7k2−50k+7<0[-10(k+1)]^2 - 4(k^2 + 1)(32) > 0 \Rightarrow 7k^2 - 50k + 7 < 0
A quadratic equation has two distinct real roots if and only if its discriminant is positive.
4
Solve the quadratic inequality 7k2−50k+7<07k^2 - 50k + 7 < 0 by factoring the quadratic expression.
17<k<7\frac{1}{7} < k < 7
The roots of (7k−1)(k−7)=0(7k - 1)(k - 7) = 0 are k=17k = \frac{1}{7} and k=7k = 7. Since the leading coefficient is positive, the expression is negative between these roots.
5
Identify and count the integer values of kk that lie within the interval (17,7)(\frac{1}{7}, 7).
The integers are 1,2,3,4,5,61, 2, 3, 4, 5, 6, which gives a total of 6 values.
The integers that are strictly greater than 17\frac{1}{7} and strictly less than 77 are the whole numbers from 11 to 66.

Key Concept

Solving systems consisting of a circle and a line algebraically by utilizing the quadratic discriminant to find the condition for two real intersection points.
Rate this question

Topics

Nonlinear Systems of Equations

More questions from this topic