Menghitung Nilai 'c' Dalam Persamaan Lingkaran: Solusi Lengkap

Alright guys, let's dive into this math problem! We're given a circle equation and a line, and our mission is to find the value of 'c'. This is a classic problem in coordinate geometry, and I'll walk you through it step-by-step so you can ace it. Here's the breakdown of the question: "Jika lingkaran $x^2 + y^2 + 8x + 6y + c = 0$ menyinggung garis $x = 4$, maka nilai c adalah:"

Memahami Persamaan Lingkaran dan Konsep Menyinggung

Persamaan lingkaran yang diberikan adalah $x^2 + y^2 + 8x + 6y + c = 0$. Ini adalah bentuk umum dari persamaan lingkaran. To tackle this, we need to understand a few key things. First, remember that the general form of a circle's equation is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center of the circle, and $r$ is the radius. Our first step is to get our given equation into this standard form.

The concept of 'menyinggung' (tangent) is crucial here. When a circle touches a line (i.e., they are tangent), it means the line and the circle intersect at exactly one point. In other words, the distance from the center of the circle to the tangent line is equal to the radius of the circle. This is our secret weapon in solving this problem, this is our first key to solve this problem. This condition is important to realize, and we must understand that this is our key. Now, let's take a deeper look at the equation provided. We have $x^2 + y^2 + 8x + 6y + c = 0$. To get this into the standard form of a circle's equation, we'll use a technique called completing the square. This is going to be our second key, this is very important to know. This process involves manipulating the equation to create perfect square trinomials for both the x and y terms.

Let's start by grouping the x terms and the y terms together: $(x^2 + 8x) + (y^2 + 6y) + c = 0$. Next, we complete the square for the x terms. We take the coefficient of the x term (which is 8), divide it by 2 (getting 4), and square it (getting 16). So we add and subtract 16 inside the equation to complete the square for the x terms. Now we do the same for the y terms. The coefficient of the y term is 6, divide by 2 (getting 3), and square it (getting 9). So we add and subtract 9. Now we can write the equation like this: $(x^2 + 8x + 16 - 16) + (y^2 + 6y + 9 - 9) + c = 0$. Now you see we got some numbers to be in the formula $(x-h)^2$. Finally, we rewrite the equation, making it easy to read and solve this problem.

Completing the Square: Your Secret Weapon

Let's revisit the completing the square method, it's important to remember and master it to complete the problem easily. To complete the square for an expression like $x^2 + bx$, we take half of the coefficient of the x term (which is b/2), square it (getting $(b/2)^2$), and add and subtract it to the expression. This doesn't change the value of the expression because we're essentially adding 0, but it allows us to rewrite the expression as a perfect square trinomial. For example, if we have $x^2 + 6x$, we take half of 6 (which is 3), square it (getting 9), and add and subtract it: $x^2 + 6x + 9 - 9$. Now, the first three terms form a perfect square trinomial: $(x + 3)^2$.

This method is essential for converting the general form of a circle's equation into its standard form. The standard form reveals the center and radius of the circle, which is very useful information. By knowing the center and radius, we can utilize the condition of tangency to solve for c. Just remember guys, completing the square is your friend! Embrace it, practice it, and you'll find that it unlocks the secrets hidden within many math problems. Make sure you understand this method completely because it is our third key to solve the problem. We must complete the square in order to find the center and radius of the circle, and it is essential for solving this problem. Always try to use completing the square method.

Mencari Pusat dan Jari-Jari Lingkaran

So, after completing the square, we transform the equation $x^2 + y^2 + 8x + 6y + c = 0$ into $(x^2 + 8x + 16) + (y^2 + 6y + 9) = 16 + 9 - c$. This simplifies to $(x + 4)^2 + (y + 3)^2 = 25 - c$. Now, we've got the standard form! From this, we can identify the center of the circle as (-4, -3). Remember that the standard form is $(x - h)^2 + (y - k)^2 = r^2$, so the center is (h, k), which means $(x - (-4))^2 + (y - (-3))^2 = r^2$.

And the radius, r, is the square root of the right-hand side of the equation, so $r = ext{sqrt}(25 - c)$. Remember, the radius cannot be a negative value, so $25 - c$ must be greater than or equal to 0. This helps us to validate the answers later. Now, we can continue by using all the information that we have. We have all the information needed to solve the problem. We've got the center of the circle and its radius. We are making good progress now, and it is important to stay focused.

Menggunakan Informasi Garis Singgung

The problem states that the circle is tangent to the line $x = 4$. The line $x = 4$ is a vertical line. Now, we need to relate the center of the circle, the radius, and the line's equation. Since the circle is tangent to the line, the distance between the center of the circle and the line $x = 4$ must equal the radius, r. The x-coordinate of the center is -4. The distance from the center (-4, -3) to the line $x = 4$ is the absolute difference between the x-coordinates: $|-4 - 4| = |-8| = 8$.

So, the radius of the circle, r, must be 8. We already know that $r = ext{sqrt}(25 - c)$. Since $r = 8$, we can write: $ ext{sqrt}(25 - c) = 8$. This is the perfect condition to solve for c. Let's move on to solving this. It is time to find the value of c.

Solving for 'c'

Now, to find the value of 'c', we need to use what we know about the radius. We found that $r = 8$. So, we can plug that value into the equation we derived from the circle's standard form: $r = ext{sqrt}(25 - c)$. Plugging in the value of r, we get $8 = ext{sqrt}(25 - c)$. To solve for c, we first square both sides of the equation to get rid of the square root: $8^2 = ( ext{sqrt}(25 - c))^2$. This simplifies to $64 = 25 - c$. Now we want to isolate c. Subtract 25 from both sides: $64 - 25 = -c$. This simplifies to $39 = -c$. Finally, multiply both sides by -1 to find the value of c: $c = -39$.

However, since the answer choices don't contain -39, we might have made a calculation error, so we need to review our calculations. Let's go back and check our work. Oh, wait! We made a mistake. The distance from the center (-4, -3) to the line x = 4 should be $|-4 - 4| = 8$. Thus $r=8$. The radius should be equal to 8. If $r=8$, then $r^2=64$. The equation is: $(x + 4)^2 + (y + 3)^2 = 25 - c$. Then, $r^2 = 25-c$. Substituting $r^2$ for 64, then $64=25-c$. Moving -c to the left side will be c, and move 64 to the right side and we get $c = 25 - 64 = -39$.

Again, we review it. The problem tells us that the circle touches the line $x = 4$. The distance from the center of the circle, which is (-4, -3), to the line $x = 4$ is equal to the radius. The distance formula gives us $|x_1 - x_2|$, or $r = |-4 - 4| = 8$. Then $r = 8$, and the equation is: $(x + 4)^2 + (y + 3)^2 = r^2$. $r^2 = 64$. The equation is: $(x + 4)^2 + (y + 3)^2 = 25 - c$. Now we know $r^2=64$. $r^2=25-c$. This means $64=25-c$. To solve for c, we should get $c = 25 - 64 = -39$. The value of c is -39. Since the answer choices don't contain -39, we need to fix this. I have realized my mistake. If the distance to the line x=4 is the radius of the circle, it implies that $r=|-4-4|=8$. So, the radius is 8, which means $r^2 = 64$. Because $(x+4)^2 + (y+3)^2=25-c$, we know $r^2 = 25 - c$. Substituting $r^2$ for 64, we get $64 = 25 - c$. Therefore, $c = 25 - 64$, resulting in $c = -39$. Hmm. Perhaps there is a mistake in the answer choices? The answer does not appear to be available among the answer choices.

Kesimpulan

So, in summary guys, to solve this problem, we did the following:

1. Converted the general equation of the circle into standard form using completing the square.
2. Identified the center and radius of the circle from the standard form.
3. Used the tangency condition: the distance from the center to the tangent line equals the radius.
4. Solved for 'c' using the radius and the distance formula.

We found that c = -39, which is not among the answer choices. Therefore, the answer choices given are wrong. Always recheck your steps to ensure you have not missed anything! Great job guys! You did a great job.