Math Problem Statement

The equation of the curve is y = (2k - 3)x^2 - kx - (k - 2), and the line y = 3x - 4 is tangent to the curve. Find the value of k.

Solution

Let's break down and solve the problem step by step from the image:

Given:

The equation of the curve is: y=(2k3)x2kx(k2)y = (2k - 3)x^2 - kx - (k - 2) where kk is a constant.

The line given is: y=3x4y = 3x - 4 which is tangent to the curve. We are tasked with finding the value of kk.

Steps to solve:

  1. Set the two equations equal to each other: Since the line is tangent to the curve, their equations will intersect at exactly one point. Set the equations equal: (2k3)x2kx(k2)=3x4(2k - 3)x^2 - kx - (k - 2) = 3x - 4

  2. Simplify to quadratic form: Rearranging the equation: (2k3)x2kx(k2)3x+4=0(2k - 3)x^2 - kx - (k - 2) - 3x + 4 = 0 Simplify the terms: (2k3)x2(k+3)x(k24)=0(2k - 3)x^2 - (k + 3)x - (k - 2 - 4) = 0 (2k3)x2(k+3)x(k6)=0(2k - 3)x^2 - (k + 3)x - (k - 6) = 0 This is a quadratic equation in the form ax2+bx+c=0ax^2 + bx + c = 0.

  3. Set the discriminant equal to zero: Since the line is tangent, the discriminant of the quadratic equation must be zero for the quadratic to have exactly one solution. The discriminant is given by: Δ=b24ac\Delta = b^2 - 4ac For the quadratic equation (2k3)x2(k+3)x(k6)=0(2k - 3)x^2 - (k + 3)x - (k - 6) = 0, the coefficients are: a=2k3,b=(k+3),c=(k6)a = 2k - 3, \quad b = -(k + 3), \quad c = -(k - 6) Setting the discriminant to zero: (k+3)24(2k3)((k6))=0(k + 3)^2 - 4(2k - 3)(-(k - 6)) = 0

  4. Solve the discriminant for kk: Expanding both terms: (k+3)2=k2+6k+9(k + 3)^2 = k^2 + 6k + 9 4(2k3)((k6))=4(2k3)(k6)-4(2k - 3)(-(k - 6)) = 4(2k - 3)(k - 6) Expand the second term: 4(2k3)(k6)=4(2k212k3k+18)=4(2k215k+18)=8k260k+724(2k - 3)(k - 6) = 4 \left(2k^2 - 12k - 3k + 18\right) = 4(2k^2 - 15k + 18) = 8k^2 - 60k + 72 Now substitute into the discriminant equation: k2+6k+9(8k260k+72)=0k^2 + 6k + 9 - (8k^2 - 60k + 72) = 0 Simplify: k2+6k+98k2+60k72=0k^2 + 6k + 9 - 8k^2 + 60k - 72 = 0 7k2+66k63=0-7k^2 + 66k - 63 = 0 Divide through by -7: k29k+9=0k^2 - 9k + 9 = 0 Factorize: (k3)2=0(k - 3)^2 = 0 So, k=3k = 3.

Thus, the value of kk is 3.

Would you like any further details or explanations on any part of this solution?

Here are 5 related questions to expand understanding:

  1. What happens if the line is not tangent but intersects the curve at two points?
  2. How do you calculate the discriminant of a quadratic equation in general?
  3. What is the geometric significance of setting the discriminant equal to zero?
  4. How do different values of kk affect the shape of the parabola given by the equation of the curve?
  5. What if the curve was a cubic function? How would you find the value of kk?

Tip:

Always check the discriminant when dealing with tangency conditions, as it provides key insights into the number of intersections.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Discriminant
Tangency Conditions

Formulas

Quadratic equation: ax^2 + bx + c = 0
Discriminant: Δ = b^2 - 4ac

Theorems

Discriminant theorem for quadratic equations

Suitable Grade Level

High School (Grades 10-12)