line with 3x-y-9=0 does not intersect with the circle x^2+px+y^2+4y=20, show that 42-root(10)<p<42+10root(10)

We are given the following line and circle equations:

1. **Line equation**: \( 3x - y - 9 = 0 \)
2. **Circle equation**: \( x^2 + px + y^2 + 4y = 20 \)

### Step 1: Express \( y \) from the line equation.
From the line equation \( 3x - y - 9 = 0 \), we can solve for \( y \):

\[
y = 3x - 9
\]

### Step 2: Substitute \( y = 3x - 9 \) into the circle equation.
Substitute this expression for \( y \) into the circle equation \( x^2 + px + y^2 + 4y = 20 \):

\[
x^2 + px + (3x - 9)^2 + 4(3x - 9) = 20
\]

Expand \( (3x - 9)^2 \) and \( 4(3x - 9) \):

\[
(3x - 9)^2 = 9x^2 - 54x + 81
\]
\[
4(3x - 9) = 12x - 36
\]

Thus, the circle equation becomes:

\[
x^2 + px + 9x^2 - 54x + 81 + 12x - 36 = 20
\]

Simplify the equation:

\[
10x^2 + (p - 42)x + 45 = 0
\]

### Step 3: Condition for no intersection.
For the line not to intersect with the circle, the quadratic equation \( 10x^2 + (p - 42)x + 45 = 0 \) must have no real solutions. This occurs when the discriminant \( \Delta \) is less than 0. The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

\[
\Delta = b^2 - 4ac
\]

Here, \( a = 10 \), \( b = p - 42 \), and \( c = 45 \). So, the discriminant is:

\[
\Delta = (p - 42)^2 - 4(10)(45)
\]

Simplify:

\[
\Delta = (p - 42)^2 - 1800
\]

For no intersection, we require \( \Delta < 0 \):

\[
(p - 42)^2 < 1800
\]

Taking the square root of both sides:

\[
|p - 42| < \sqrt{1800}
\]

\[
|p - 42| < 42.426
\]

Thus, the range for \( p \) is:

\[
42 - \sqrt{1800} < p < 42 + \sqrt{1800}
\]

Since \( \sqrt{1800} = \sqrt{10 \times 180} = \sqrt{10} \times \sqrt{180} \approx 42.426 \), this gives us the range:

\[
42 - \sqrt{10} < p < 42 + \sqrt{10}
\]

This shows that the value \( p \) lies within that

Explore how to determine the range of p for which the line 3x - y - 9 = 0 does not intersect with the circle x^2 + px + y^2 + 4y = 20. This problem involves key algebraic concepts, including quadratic equations and the discriminant, and is suitable for students in Grades 10-12.

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