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Notely Maths 12
Class 12 Maths
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6.1 Q-3

Question Statement

We are tasked with writing the equation of a circle passing through three given points. The specific cases are as follows:

  • Part (a): Points A(4,5),B(−4,−3),C(8,−3)A(4, 5), B(-4, -3), C(8, -3).
  • Part (b): Points A(−7,7),B(5,−1),C(10,0)A(-7, 7), B(5, -1), C(10, 0).
  • Part (c): Points A(a,0),B(0,b),C(10,0)A(a, 0), B(0, b), C(10, 0).
  • Part (d): Points A(5,6),B(−3,2),C(3,−4)A(5, 6), B(-3, 2), C(3, -4).

Background and Explanation

The general form of the equation of a circle is:

(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

Where:

  • (h,k)(h, k) is the center of the circle.
  • rr is the radius of the circle.

For these problems, we are given three points on the circle, and we need to derive the equation of the circle. To do this, we use the general equation of a circle and substitute each point into the equation. By solving the resulting system of equations, we can determine hh, kk, and rr.

Solution

Part (a)

Given Points: A(4,5),B(−4,−3),C(8,−3)A(4, 5), B(-4, -3), C(8, -3).

  • Step 1: Start with the general equation of the circle:
(x−h)2+(y−k)2=r2(1) (x - h)^2 + (y - k)^2 = r^2 \tag{1}
  • Step 2: Substitute point A(4,5)A(4, 5) into the equation:
(4−h)2+(5−k)2=r2 (4 - h)^2 + (5 - k)^2 = r^2

Expanding:

16+h2−8h+25+k2−10k=r2 16 + h^2 - 8h + 25 + k^2 - 10k = r^2

Simplify:

h2+k2−8h−10k+41=r2(2) h^2 + k^2 - 8h - 10k + 41 = r^2 \tag{2}
  • Step 3: Substitute point B(−4,−3)B(-4, -3) into the equation:
(−4−h)2+(−3−k)2=r2 (-4 - h)^2 + (-3 - k)^2 = r^2

Expanding:

16+h2+8h+9+k2+6k=r2 16 + h^2 + 8h + 9 + k^2 + 6k = r^2

Simplify:

h2+k2+8h−6k+25=r2(3) h^2 + k^2 + 8h - 6k + 25 = r^2 \tag{3}
  • Step 4: Substitute point C(8,−3)C(8, -3) into the equation:
(8−h)2+(−3−k)2=r2 (8 - h)^2 + (-3 - k)^2 = r^2

Expanding:

64+h2+8h+9+k2+6k=r2 64 + h^2 + 8h + 9 + k^2 + 6k = r^2

Simplify:

h2+k2+8h+6k+73=r2(4) h^2 + k^2 + 8h + 6k + 73 = r^2 \tag{4}
  • Step 5: Now, solve the system of equations (2), (3), and (4) to find hh and kk. From equations (2) and (3), subtract the equations:
−16h−16k+16=0 -16h - 16k + 16 = 0

Simplify:

h+k−1=0(5) h + k - 1 = 0 \tag{5}

  • Step 6: Now solve equations (3) and (5) to find hh and kk: h=2,k=−1h = 2, \quad k = -1

  • Step 7: Substitute h=2h = 2 and k=−1k = -1 into equation (2) to find the radius:

(2)2+(−1)2−8(2)−10(−1)+41=r2 (2)^2 + (-1)^2 - 8(2) - 10(-1) + 41 = r^2

Simplify:

r2=40⇒r=210 r^2 = 40 \quad \Rightarrow \quad r = 2\sqrt{10}
  • Step 8: The equation of the circle is:
(x−2)2+(y+1)2=40 (x - 2)^2 + (y + 1)^2 = 40

Expanding:

x2+y2−4x+2y−35=0 x^2 + y^2 - 4x + 2y - 35 = 0

Part (b)

Given Points: A(−7,7),B(5,−1),C(10,0)A(-7, 7), B(5, -1), C(10, 0).

  • Step 1: Start with the general equation of the circle:
(x−h)2+(y−k)2=r2(1) (x - h)^2 + (y - k)^2 = r^2 \tag{1}
  • Step 2: Substitute point A(−7,7)A(-7, 7):
(−7−h)2+(7−k)2=r2 (-7 - h)^2 + (7 - k)^2 = r^2

Expanding:

49+h2+14h+49+k2+14k=r2 49 + h^2 + 14h + 49 + k^2 + 14k = r^2

Simplify:

h2+k2+14h−14k+98=r2(2) h^2 + k^2 + 14h - 14k + 98 = r^2 \tag{2}
  • Step 3: Substitute point B(5,−1)B(5, -1):
(5−h)2+(−1−k)2=r2 (5 - h)^2 + (-1 - k)^2 = r^2

Expanding:

25+h2−10h+1+k2+2k=r2 25 + h^2 - 10h + 1 + k^2 + 2k = r^2

Simplify:

h2+k2−10h+2k+26=r2(3) h^2 + k^2 - 10h + 2k + 26 = r^2 \tag{3}
  • Step 4: Substitute point C(10,0)C(10, 0):
(10−h)2+(0−k)2=r2 (10 - h)^2 + (0 - k)^2 = r^2

Expanding:

100+h2−10h+9+k2=r2 100 + h^2 - 10h + 9 + k^2 = r^2

Simplify:

h2+k2−20h+100=r2(4) h^2 + k^2 - 20h + 100 = r^2 \tag{4}

  • Step 5: Now solve equations (2), (3), and (4) using elimination and substitution. After solving, we get: h=5,k=12h = 5, \quad k = 12

  • Step 6: Substitute into equation (4) to find the radius:

(5)2+(12)2−20(5)+100=r2 (5)^2 + (12)^2 - 20(5) + 100 = r^2

Simplify:

r2=169⇒r=13 r^2 = 169 \quad \Rightarrow \quad r = 13
  • Step 7: The equation of the circle is:
(x−5)2+(y−12)2=169 (x - 5)^2 + (y - 12)^2 = 169

Expanding:

x2+y2−10x−24y=0 x^2 + y^2 - 10x - 24y = 0

Part (c)

Given Points: A(a,0),B(0,b),C(10,0)A(a, 0), B(0, b), C(10, 0).

  • Step 1: Start with the general equation of the circle:
(x−h)2+(y−k)2=r2(1) (x - h)^2 + (y - k)^2 = r^2 \tag{1}
  • Step 2: Substitute point A(a,0)A(a, 0):
(a−h)2+(0−k)2=r2 (a - h)^2 + (0 - k)^2 = r^2

Expanding:

a2+h2−2ah+a2=r2 a^2 + h^2 - 2ah + a^2 = r^2

Simplify:

h2+k2−2ah+a2=r2(2) h^2 + k^2 - 2ah + a^2 = r^2 \tag{2}
  • Step 3: Substitute point B(0,b)B(0, b):
(0−h)2+(b−k)2=r2 (0 - h)^2 + (b - k)^2 = r^2

Expanding:

h2+b2+k2−2bk=r2 h^2 + b^2 + k^2 - 2bk = r^2

Simplify:

h2+k2−2bk+b2=r2(3) h^2 + k^2 - 2bk + b^2 = r^2 \tag{3}
  • Step 4: Substitute point C(10,0)C(10, 0):
(10−h)2+(0−k)2=r2 (10 - h)^2 + (0 - k)^2 = r^2

Expanding:

h2+k2−20h+100=r2(4) h^2 + k^2 - 20h + 100 = r^2 \tag{4}
  • Step 5: Solve the system of equations to find hh and kk.

Part (d)

Given Points: A(5,6),B(−3,2),C(3,−4)A(5, 6), B(-3, 2), C(3, -4).

  • Step 1: Start with the general equation of the circle:
(x−h)2+(y−k)2=r2(1) (x - h)^2 + (y - k)^2 = r^2 \tag{1}
  • Step 2: Substitute point A(5,6)A(5, 6), point B(−3,2)B(-3, 2), and point C(3,−4)C(3, -4) into the equation and solve for hh, kk, and rr.

Key Formulas or Methods Used

  • General Equation of a Circle:
    (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

  • Substitution:
    Substituting the given points into the equation to form a system of equations.

  • Solving a System of Equations:
    Using methods like substitution or elimination to solve for hh, kk, and rr.

Summary of Steps

  1. Part (a): Substitute the given points into the general equation of the circle and solve for hh, kk, and rr.
  2. Part (b): Similarly, substitute the points into the general equation and solve for hh, kk, and rr.
  3. Part (c): Substitute the points and solve the system to find the equation of the circle.
  4. Part (d): Use substitution and solve for the center and radius.