Answer each question. Does the straight line 3x - 2y = 9 intersec...

Question

Answer each question. Does the straight line 3x - 2y = 9 intersect the circle x^2 + y^2 = 25? (Hint: To find out, solve the system formed by these two equations.)

Accepted Answer

Determine whether the straight line six X minus five, Y equals 11 intersects the given circle with our circle being X squared plus Y squared equals 49. Now, let's solve this. We want to see if our intersection has a real or a non real answer to do that. We're gonna substitute our straight line equation into our circle equation. Let's first solve our straight line equation for X. It's six X minus five Y equals 11. Solve for X by adding five Y to both sets and dividing by six divided by six, we get X equals five, Y plus 11 divided by six. No, we can take Rx and plug it into the circle equation. We get five Y plus 11 divided by six squared plus Y squared equals 49. Now, we can square our fraction. You get five Y plus 11 square divided by six squared, which is 36. Now, we can add the Y squared and we can add it by having the same denominator as our first fraction. Let's say we change our Y squared to 36 Y squared divided by 36. We can now expand our polynomial five Y plus 11 squared. This expands to 25 Y square plus 1 10, Y plus 1 21. We'll also add our 36 white square and put it all on top of the fraction with 36 in the denominator. This equals 49. Let's combine all of our things. We can multiply our 36 to the other side to get rid of on the left side. This equals 17 64 on the left side 24 Y squared plus 36 Y squared gives us 61 Y squared. Add 1 10 Y and 1 subtract 17 64 from both sides to set it equals to zero. We get 61 Y squared plus 1 10, Y minus 16 43. This equals zero. Now that we have our quadratic, we can use quadratic formula to find our solutions. Let's say Y equals negative B. So you have negative 1 10 plus or minus the square root of B squared, which is 1 10 squared minus four, multiplied by a 61 multiple by C of 43. All divided by two, multiplied by A which is 61 which in our quadratic, we know A is our first term. B is our second term and C is our final term of our quadratic. Now, we can simplify Y equals negative 1 plus or minus the square root of 12, 100 plus 400,892. When simplified. This is all divided by 1 by equals negative 1 10 plus or minus square root of 412, divided by 1 22. Now we notice here our solutions for Y will be real answers because our square root has a positive number underneath it. If we were to evaluate this, we would get a real answer. Which means we also have a real answer for X values. So both our X and Y are real, which means our line does indeed intersect our circle. We will then say the answer to our problem is answer a yes. Our line intersects the circle. OK. I hope to help you solve the problem. Thank you for watching. Goodbye.

Answer each question. Does the straight line 3x - 2y = 9 intersect the circle x^2 + y^2 = 25? (Hint: To find out, solve the system formed by these two equations.)

Key Concepts

Linear Equations

Circle Equation

Systems of Equations

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