Math Problem Statement

example on how to solve:

0:00

  • WE ARE ASKED TO WRITE THE CENTERED EQUATION OF A CIRCLE 0:03 WITH A DIAMETER WITH END POINTS (-2,1) AND (4,3). 0:07 THE CENTERED EQUATION OF A CIRCLE IS GIVEN HERE 0:10 WHERE (H,K) ARE THE COORDINATES OF THE CENTER, 0:14 AND R IS THE LENGTH OF THE RADIUS. 0:17 SO KNOWING THE TWO ENDPOINTS OF THE DIAMETER 0:19 WHICH IS A SEGMENT THAT PASSES THROUGH THE CENTER OF THE CIRCLE 0:23 AND HAS TWO END POINTS ON THE CIRCLE, 0:25 WE WANT TO FIND THE CENTER OF THE CIRCLE 0:27 AND THE LENGTH OF THE RADIUS. 0:29 TO UNDERSTAND HOW TO APPROACH THIS, 0:31 LET'S TAKE A LOOK AT THE CIRCLE THAT HAS THESE TWO ENDPOINTS 0:34 FOR A DIAMETER. 0:36 HERE'S THE DIAMETER WE WERE REFERRING TO. 0:37 NOTICE HOW IT HAS ONE ENDPOINT AT (-2,1), 0:41 AND ANOTHER ENDPOINT AT (4,3). 0:43 AGAIN, OUR GOAL HERE IS TO FIND THE COORDINATES OF THE CENTER 0:47 WHICH WOULD BE (H,K), AND THEN THE LENGTH OF THE RADIUS 0:53 WHICH WOULD BE THE LENGTH OF THIS SEGMENT HERE. 0:57 NOTICE HOW THE CENTER OF THE CIRCLE WOULD 1:00 BE THE MIDPOINT OF THE DIAMETER. 1:02 AND THEN ONCE WE FIND THE MIDPOINT 1:03 OR THE CENTER OF THE CIRCLE, 1:05 WE CAN THEN FIND THE LENGTH OF THE RADIUS 1:07 BY DETERMINING THE DISTANCE FROM THE CENTER 1:09 TO ONE OF THE TWO POINTS ON THE CIRCLE. 1:12 SO LET'S GO AHEAD AND DO THAT. 1:14 AND, AGAIN, THE CENTER WOULD HAVE COORDINATES (H,K) 1:19 WHICH WOULD BE THE MIDPOINT OF THE DIAMETER. 1:22 THE MIDPOINT FORMULA IS GIVEN HERE FOR REVIEW. 1:25 WE'RE BASICALLY GOING TO FIND THE AVERAGE OF THE X COORDINATES 1:28 AND THE AVERAGE OF THE Y COORDINATES. 1:33 LET'S CALL THESE COORDINATES THE 1s, X SUB 1 AND Y SUB 1. 1:37 AND THESE COORDINATES THE 2, X SUB 2, Y SUB 2. 1:42 SO WE WOULD HAVE -2 + 4 DIVIDED BY 2 AND 1 + 3 DIVIDED BY 2. 1:54 WELL, -2 + 4 = 2. 1:56 2 DIVIDED BY 2 IS 1. 1:59 AND 1 + 3 = 4. 2:00 4 DIVIDED BY 2 IS 2. 2:05 SO NOW WE KNOW THAT H = 1 AND K = 2. 2:13 AND NOW TO FIND THE LENGTH OF THE RADIUS 2:14 WE'LL FIND THE DISTANCE FROM THE CENTER 2:17 TO ONE OF THE ENDPOINTS OF THE DIAMETER. 2:20 SO LET'S GO AHEAD AND USE THE CENTER 2:25 AND THIS FIRST ENDPOINT (-2,1). 2:33 SO THE LENGTH OF THE RADIUS, OR R, IS GOING TO BE EQUAL 2:36 TO THE DISTANCE BETWEEN THESE TWO POINTS 2:41 AND THE DISTANCE FORMULA IS GIVEN HERE. 2:44 SO LET'S GO AHEAD AND CALL THESE THE 1s, 2:46 X SUB 1 AND Y SUB 1, 2:47 AND THESE THE 2s, X SUB 2, Y SUB 2. 2:50 SO THE LENGTH OF THE RADIUS OR THE DISTANCE 2:52 IS GOING TO BE EQUAL TO X SUB 2 - X SUB 1, 2:57 OR -2 - 1 SQUARED + Y SUB 2 - Y SUB 1 OR 1 - 2 SQUARED. 3:07 SO WE'LL HAVE THE SQUARE ROOT OF -3 SQUARED. 3:11 THAT'S 9 PLUS (1 - 2) = -1. 3:15 -1 SQUARED IS 1. 3:17 SO THE RADIUS IS EQUAL TO THE SQUARE ROOT 10. 3:22 BUT FOR THE STANDARD EQUATION OF THE CIRCLE 3:24 WE NEED R SQUARED, NOT R. 3:27 SO IF WE KNOW R EQUALS SQUARE ROOT 10, 3:30 WE CAN FIND R SQUARED 3:31 BY SQUARING BOTH SIDES OF THE EQUATION. 3:38 AND THE SQUARE ROOT OF 10 SQUARED WOULD JUST BE 10. 3:41 NOW THAT WE HAVE (H,K) AND R SQUARED 3:48 WE CAN WRITE THE EQUATION IN STANDARD FORM. 3:52 WE WOULD HAVE THE QUANTITY X - H WHERE H IS 1. 3:57 SO WE HAVE X - 1 SQUARED + (Y - K) WHERE K IS 2. 4:04 SO WE HAVE Y - 2 SQUARED = R SQUARED WHERE WE KNOW R = 10. 4:15 THIS WOULD BE THE STANDARD EQUATION OF THE CIRCLE 4:18 WITH THE DIAMETER WITH THE GIVEN ENDPOINTS. 4:22 I HOPE YOU FOUND THIS EXPLANATION HELPFUL.
Find the standard form for the equation of a circle (x - h)² + (y - k)² = r² with a diameter that has endpoints (−1, −6) and (0, −3).

Solution

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image_path = "/mnt/data/1732222683956lckrn49k.jpg" text = pytesseract.image_to_string(Image.open(image_path))

text'Find the standard form for the equation of a circle\n(x —h)?+(y—kP =r?\n\nwith a diameter that has endpoints (—1, —6) and (0, —3).\n\nh =| -0.5 v| & -0.5\nk =| -4.5 v| & -45\nr=|| |\n\nQuestion Help: ©) Video\n\nWeenie lelSaleyay) §~Jump to Answer\n''Find the standard form for the equation of a circle\n(x —h)?+(y—kP =r?\n\nwith a diameter that has endpoints (—1, —6) and (0, —3).\n\nh =| -0.5 v| & -0.5\nk =| -4.5 v| & -45\nr=|| |\n\nQuestion Help: ©) Video\n\nWeenie lelSaleyay) §~Jump to Answer\n'successThe problem asks to find the standard form of the equation of a circle with a given diameter having endpoints (−1,−6)(-1, -6) and (0,−3)(0, -3).

Solution Steps:

  1. Find the Center (h,k)(h, k): The center of the circle is the midpoint of the diameter. Using the midpoint formula: h=x1+x22,k=y1+y22h = \frac{x_1 + x_2}{2}, \quad k = \frac{y_1 + y_2}{2} Substituting: h=−1+02=−0.5,k=−6+(−3)2=−4.5h = \frac{-1 + 0}{2} = -0.5, \quad k = \frac{-6 + (-3)}{2} = -4.5

    So, the center is (−0.5,−4.5)(-0.5, -4.5).

  2. Find the Radius rr: The radius is half the length of the diameter. We calculate the length of the diameter using the distance formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} Substituting: d=(0−(−1))2+((−3)−(−6))2=12+32=1+9=10d = \sqrt{(0 - (-1))^2 + ((-3) - (-6))^2} = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10} The radius is: r=d2=102r = \frac{d}{2} = \frac{\sqrt{10}}{2} In terms of r2r^2 (used in the equation), we have: r2=(102)2=104=2.5r^2 = \left(\frac{\sqrt{10}}{2}\right)^2 = \frac{10}{4} = 2.5

  3. Write the Standard Equation: The equation of the circle in standard form is: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2 Substituting h=−0.5h = -0.5, k=−4.5k = -4.5, and r2=2.5r^2 = 2.5: (x+0.5)2+(y+4.5)2=2.5(x + 0.5)^2 + (y + 4.5)^2 = 2.5

Final Answer:

The standard equation of the circle is: (x+0.5)2+(y+4.5)2=2.5(x + 0.5)^2 + (y + 4.5)^2 = 2.5

If you'd like further clarification or steps, let me know! Here are some additional questions to explore:

  1. How do you derive the midpoint formula?
  2. Can you explain the distance formula's geometric significance?
  3. How does the standard form of the circle equation differ from general form?
  4. What does r2r^2 represent geometrically?
  5. How would this process differ if you were given the radius instead of the diameter?

Tip: Always double-check calculations involving fractions and square roots to ensure accuracy!

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Math Problem Analysis

Mathematical Concepts

Geometry
Equations of Circles
Midpoint Formula
Distance Formula

Formulas

Midpoint Formula: (h, k) = ((x1 + x2)/2, (y1 + y2)/2)
Distance Formula: d = √((x2 - x1)² + (y2 - y1)²)
Standard Form of Circle: (x - h)² + (y - k)² = r²

Theorems

-

Suitable Grade Level

Grades 9-10

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