Question 10 of 70intermediate🔧 ApplyLong Answer4 marks

Find the equation of the circle passing through the points (4, 1) and (6, 5) and whose centre is on the line 4x + y = 16.

Correct Answer

Let the equation of the circle be (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, where (h,k)(h, k) is the centre and rr is the radius.

Since the centre (h,k)(h, k) lies on the line 4x+y=164x + y = 16, we have: Equation for centre: 4h+k=164h + k = 16.

Since the circle passes through (4,1)(4, 1) and (6,5)(6, 5), these points satisfy the circle's equation. Equating the distances (radii) from the centre to these points: (4−h)2+(1−k)2=(6−h)2+(5−k)2(4 - h)^2 + (1 - k)^2 = (6 - h)^2 + (5 - k)^2.

Expanding and simplifying:

16−8h+h2+1−2k+k2=36−12h+h2+25−10k+k216 - 8h + h^2 + 1 - 2k + k^2 = 36 - 12h + h^2 + 25 - 10k + k^2 17−8h−2k=61−12h−10k17 - 8h - 2k = 61 - 12h - 10k 4h+8k=444h + 8k = 44

Simplified relation: h+2k=11h + 2k = 11.

Solving the system of linear equations 4h+k=164h + k = 16 and h+2k=11h + 2k = 11: From h+2k=11h + 2k = 11, we get h=11−2kh = 11 - 2k. Substituting in the first equation: 4(11−2k)+k=164(11 - 2k) + k = 16

44−8k+k=1644 - 8k + k = 16

−7k=−28  ⟹  k=4-7k = -28 \implies k = 4. Then, h=11−2(4)=3h = 11 - 2(4) = 3. Centre: (h,k)=(3,4)(h, k) = (3, 4).

Finally, calculating the radius squared using point (4,1)(4, 1): r2=(4−3)2+(1−4)2=12+(−3)2=1+9=10r^2 = (4 - 3)^2 + (1 - 4)^2 = 1^2 + (-3)^2 = 1 + 9 = 10.

Hence, the equation of the required circle is: (x−3)2+(y−4)2=10(x - 3)^2 + (y - 4)^2 = 10

Exercise: EXERCISE 10.1 | Q: 10 | (Page 6)
For More Understanding

Explanation

This solution follows the methodology shown in Example 4 of the provided text. The general equation of a circle is established, and conditions are applied: substituting the center coordinates into the line equation, and equating the radius squared values derived from the two points the circle passes through. Solving the resulting system of linear equations yields the center coordinates, which are then used to find the final equation.

Solution Steps

  1. Step 1: Define the general equation of the circle as (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2.

  2. Step 2: Use the condition that the center lies on the line 4x+y=164x + y = 16 to get 4h+k=164h + k = 16.

  3. Step 3: Equate the expressions for r2r^2 obtained from points (4,1)(4, 1) and (6,5)(6, 5) to derive the linear relation h+2k=11h + 2k = 11.

  4. Step 4: Solve the system of linear equations to find the center (h,k)=(3,4)(h, k) = (3, 4).

  5. Step 5: Calculate r2=10r^2 = 10 using the center and one point, then state the final equation.

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