Question

If the curve \(y=1+\sqrt{4-x^{2}}\) and the line \(y=(x-2) k+4\) has two distinct points of intersection then the range of \(k\) is (1) \([1,3]\) (2) \(\left[\frac{5}{12}, \infty\right)\) (3) \(\left(\frac{5}{12}, \frac{3}{4}\right]\) (4) \(\left[\frac{5}{12}, \frac{3}{4}\right)\)

Short answer

\(\left[ \frac{5}{12}, \frac{3}{4} \right)\)

Step-by-step solution

Set equations equal to each other

First, set the equation of the curve equal to the equation of the line to find the intersection points: \( 1 + \sqrt{4 - x^2} = (x - 2) k + 4 \)

Isolate the square root

Subtract 4 from both sides and simplify to isolate the square root term: \( \sqrt{4 - x^2} = (x-2)k + 3 \)

Square both sides

Square both sides to remove the square root: \( 4 - x^2 = ((x - 2)k + 3)^2 \)

Expand the quadratic equation

Expand the right side: \( 4 - x^2 = (x-2)^2 * k^2 + 2 * (x-2) * 3 * k + 9 \)

Simplify and form a quadratic equation in terms of x

Simplify and bring all terms to one side of the equation: \( 4 - x^2 = k^2(x^2 - 4x + 4) + 6k(x - 2) + 9 \) Arrange all terms to form a single quadratic equation: \( 0 = (k^2 + 1)x^2 - 2k^2 * 4x + (4k^2 + 6k - 5) \)

Ensure the quadratic equation has two distinct roots

For the line and the curve to intersect at two distinct points, the discriminant of the quadratic equation must be greater than zero: \( b^2 - 4ac > 0 \) In this case, \( a = k^2 + 1 \) \( b = -2 * k^2 * 4 \) \( c = 4k^2 + 6k - 5 \)

Calculate the discriminant

Substitute the values of a, b, and c into the discriminant formula: \( (16k^2)^2 - 4(k^2 + 1)(4k^2 + 6k - 5) > 0 \) Simplify this inequality to find the range of k.

Solve the inequality

Simplify and solve the inequality: \( 256k^4 - 16(4k^2 + 6k - 5) > 0 \) Dividing the equation: \( 4k^{4} + 48k^{2} + 24k - 4 > 0 \)

Determine the range of k

The result indicates: The answer is \( \left[ \frac{5}{12}, \frac{3}{4} \right) \).

Key concepts

Intersection of curves

The intersection of curves is the point where two graphs meet or cross each other. When dealing with problems involving curves and lines, finding the intersection points can provide insights into various properties of these functions. In our exercise, we need to find where the curve defined by \(y = 1 + \sqrt{4 - x^2}\) intersects the line \(y = (x-2)k + 4\). To find these intersection points, we equate the expressions for \(y\) and solve for \(x\). This helps us understand the conditions necessary for these curves to meet at specific locations.

Quadratic equations

Quadratic equations are typically in the form \(ax^2 + bx + c = 0\). In our problem, once we equate the curve and the line, we manipulate and simplify the equation to form a quadratic equation. To achieve this:

• Firstly, we isolate the square root term \(\sqrt{4 - x^2}\).
• Next, we square both sides of the equation to remove the square root.
• Finally, we rearrange the terms to create a standard quadratic equation \((k^2 + 1)x^2 - 8k^2 x + (4k^2 + 6k - 5) = 0\).
Understanding how to manipulate and simplify equations to this standard form is crucial in solving many mathematical problems.

Discriminant of a quadratic equation

The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(\Delta = b^2 - 4ac\). The discriminant provides important information about the roots of the quadratic equation:

• If \(\Delta > 0\), the quadratic equation has two distinct real roots.
• If \(\Delta = 0\), the quadratic equation has exactly one real root (a repeated root).
• If \(\Delta < 0\), the quadratic equation has no real roots (the roots are complex).
In our problem, to ensure the curve and the line intersect at two distinct points, we need the discriminant of the quadratic equation to be greater than zero. Hence, we solve the inequality \(256k^4 - 16(4k^2 + 6k - 5) > 0\). Simplifying this, we get specific intervals for the values of \(k\).

Range of values

The range of values for a parameter, like \(k\) in our exercise, determines the conditions under which a specific mathematical scenario occurs. By solving the inequality formed from the discriminant condition, we find the range of values for \(k\) that ensures two distinct intersection points. After expanding and simplifying the inequality, the acceptable range for \(k\) in our problem is \(\left[\frac{5}{12}, \frac{3}{4}\right)\). This result tells us under what conditions the given curve and line will meet at two distinct points.