The problem you’ve shared involves determining when a line is tangent to a circle, using the concept of the discriminant. Let's break it down:

Problem Summary:

• You are given a line equation: $2x + y - 5 = 0$ and a circle equation: $(x - 3)^2 + (y - p)^2 = 5$
• The problem is to:
  1. Find the values of $p$ such that the line is tangent to the circle.
  2. Write down the coordinates of the center of the circle in each case.

Step 1: Substituting the line equation into the circle equation

We will substitute $y = -2x + 5$ from the line into the circle equation to form a quadratic in $x$:

The circle equation is: $(x - 3)^2 + (y - p)^2 = 5$ Substitute $y = -2x + 5$: $(x - 3)^2 + (-2x + 5 - p)^2 = 5$

Step 2: Expanding and simplifying

1. Expand $(x - 3)^2$ and $(-2x + 5 - p)^2$: $(x - 3)^2 = x^2 - 6x + 9$ $(-2x + 5 - p)^2 = (4x^2 - 20x + 25 - 10p + p^2)$

2. Plug these back into the equation: $x^2 - 6x + 9 + 4x^2 - 20x + 25 - 10p + p^2 = 5$ Simplify: $5x^2 - 26x + (34 - 10p + p^2) = 5$ Subtract 5 from both sides: $5x^2 - 26x + (29 - 10p + p^2) = 0$

Step 3: Use the discriminant to find the condition for tangency

For the line to be tangent to the circle, the quadratic must have exactly one solution, which means the discriminant must be zero: $\Delta = b^2 - 4ac = 0$ Here, $a = 5$, $b = -26$, and $c = 29 - 10p + p^2$.

1. Compute the discriminant: $\Delta = (-26)^2 - 4(5)(29 - 10p + p^2)$ $\Delta = 676 - 20(29 - 10p + p^2)$ $\Delta = 676 - 580 + 200p - 20p^2$ $\Delta = 96 + 200p - 20p^2$ Set $\Delta = 0$ for tangency: $96 + 200p - 20p^2 = 0$ Divide by 4 to simplify: $24 + 50p - 5p^2 = 0$ Multiply by -1 to rearrange: $5p^2 - 50p - 24 = 0$

Step 4: Solve the quadratic equation for $p$

Solve using the quadratic formula: $p = \frac{-(-50) \pm \sqrt{(-50)^2 - 4(5)(-24)}}{2(5)}$ $p = \frac{50 \pm \sqrt{2500 + 480}}{10}$ $p = \frac{50 \pm \sqrt{2980}}{10}$ $p = \frac{50 \pm 54.6}{10}$ Thus, the two possible values of $p$ are: $p_1 = \frac{50 + 54.6}{10} = 10.46, \quad p_2 = \frac{50 - 54.6}{10} = -0.46$

Step 5: Coordinates of the center of the circle

For each value of $p$, the coordinates of the center of the circle are:

• When $p_1 = 10.46$, the center is $(3, 10.46)$
• When $p_2 = -0.46$, the center is $(3, -0.46)$

Would you like a deeper explanation of any step or any further details?

Related Questions:

1. How do you derive the equation of a line tangent to a circle in general form?
2. How does the discriminant help in determining the nature of solutions to a quadratic equation?
3. Can you solve for other values of $p$ if the line intersects the circle at two points?
4. How does changing the center of the circle affect the condition for tangency?
5. What happens to the quadratic discriminant when a line does not intersect the circle?

Tip:

Always simplify your discriminant equation as much as possible before solving; it often reduces calculation errors.