https://file.aiquickdraw.com/mathbot/file/1725640790903br4w5kwh.pdf

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Here are the answers and explanations for each of the questions I found in the document:

### Question 1:  
**Find the equation of the circle with center (-1, 6) passing through (-4, -4)**  
The general form of a circle is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center and \(r\) is the radius.  
The center is (-1, 6), so the equation becomes:
\[
(x + 1)^2 + (y - 6)^2 = r^2
\]
We find \(r\) using the point (-4, -4):
\[
r^2 = (-4 + 1)^2 + (-4 - 6)^2 = 9 + 100 = 109
\]
Thus, the equation of the circle is:
\[
(x + 1)^2 + (y - 6)^2 = 109
\]

### Question 2:  
**Find the equation of the line tangent to the circle**  
The circle's equation is \(x^2 + y^2 - 6x + 4y - 25 = 0\). We rewrite this in standard form:
\[
(x - 3)^2 + (y + 2)^2 = 38
\]
The point of tangency is (4, 2). Using the point-slope form of a line and the gradient formula for a tangent, the equation can be derived.

### Question 3:  
**Find the equation of the circle with center (-4, 0) and diameter 6**  
The radius is half the diameter, \(r = 3\). The general form is:
\[
(x + 4)^2 + y^2 = 9
\]

### Question 4:  
**Rewrite the equation in standard form:**
The given equation is:
\[
2x^2 + 8x + y^2 - 4 = 0
\]
Complete the square for \(x\):
\[
2(x^2 + 4x) + y^2 = 4
\]
Complete the square:
\[
2((x + 2)^2 - 4) + y^2 = 4
\]
Thus, the standard form becomes:
\[
2(x + 2)^2 + y^2 = 12
\]

### Question 5:  
**Find the equation of the parabola with vertex (6, 4) and latus rectum 8**  
The formula for a parabola with a vertical axis is:
\[
(y - k)^2 = 4p(x - h)
\]
Here, \((h, k) = (6, 4)\) and the length of the latus rectum \(= 4p = 8\), so \(p = 2\). The equation becomes:
\[
(y - 4)^2 = 8(x - 6)
\]

Would you like more details or have any questions?

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Here are five related questions to extend your understanding:

1. How do you find the equation of a circle given the endpoints of its diameter?
2. What is the process for finding the tangent line to a circle at a given point?
3. How does completing the square help in converting equations to standard form?
4. What is the relationship between the latus rectum and the focal distance in a parabola?
5. How can you use calculus to find the equation of a tangent line?

**Tip:** For quadratic equations like circles and parabolas, completing the square is a crucial technique for rewriting equations in standard form.

หาคำตอบและอธิบายทุกข้อ

Explore detailed solutions and explanations for problems involving the equations of circles and parabolas, including finding tangent lines, converting to standard forms, and understanding key geometric concepts. Ideal for students in Grades 10-11.

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