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Circle question

2006 · Shift 0 · Q78
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Circle question

2006 · Shift 0 · Q78

JEE MainMathematicsCircleMCQ+4 / −1
If the lines 3x−4y−7=03x - 4y - 7 = 03x−4y−7=0 and 2x−3y−5=02x - 3y - 5 = 02x−3y−5=0 are two diameters of a circle of area 49π49\pi49π square units, the equation of the circle is :
  1. A
     x2+y2+2x −2y−47=0 \,{x^2} + {y^2} + 2x\, - 2y - 47 = 0\,x2+y2+2x−2y−47=0
  2. B
     x2+y2+2x −2y−62=0 \,{x^2} + {y^2} + 2x\, - 2y - 62 = 0\,x2+y2+2x−2y−62=0
  3. C
    x2+y2−2x +2y−62=0{x^2} + {y^2} - 2x\, + 2y - 62 = 0x2+y2−2x+2y−62=0
  4. D
    x2+y2−2x +2y−47=0{x^2} + {y^2} - 2x\, + 2y - 47 = 0x2+y2−2x+2y−47=0
View written solutionFree

Correct answer: D

  1. Use the area to find the radius

Given area of the circle: πr2=49π\pi r^2 = 49\piπr2=49π So, r2=49  ⟹  r=7r^2 = 49 \implies r = 7r2=49⟹r=7

  1. Find the center of the circle

Since the given lines are diameters, both must pass through the center of the circle. Hence, the center is the point of intersection of 3x−4y−7=03x - 4y - 7 = 03x−4y−7=0 and 2x−3y−5=02x - 3y - 5 = 02x−3y−5=0

Solve the system: 3x−4y=7...(1)3x - 4y = 7 \quad ...(1)3x−4y=7...(1) 2x−3y=5...(2)2x - 3y = 5 \quad ...(2)2x−3y=5...(2)

Multiply (1) by 222: 6x−8y=146x - 8y = 146x−8y=14 Multiply (2) by 333: 6x−9y=156x - 9y = 156x−9y=15

Subtract: y=−1y = -1y=−1

Substitute into (2): 2x−3(−1)=52x - 3(-1) = 52x−3(−1)=5 2x+3=52x + 3 = 52x+3=5 2x=2  ⟹  x=12x = 2 \implies x = 12x=2⟹x=1

So the center is (1,−1)(1,-1)(1,−1)

  1. Write the equation of the circle

Standard form: (x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2(x−h)2+(y−k)2=r2 With center (1,−1)(1,-1)(1,−1) and radius 777: (x−1)2+(y+1)2=49(x-1)^2 + (y+1)^2 = 49(x−1)2+(y+1)2=49

Expand: x2−2x+1+y2+2y+1=49x^2 - 2x + 1 + y^2 + 2y + 1 = 49x2−2x+1+y2+2y+1=49 x2+y2−2x+2y+2=49x^2 + y^2 - 2x + 2y + 2 = 49x2+y2−2x+2y+2=49 x2+y2−2x+2y−47=0x^2 + y^2 - 2x + 2y - 47 = 0x2+y2−2x+2y−47=0

  1. Compare with the options

This matches:

D: x2+y2−2x+2y−47=0x^2 + y^2 - 2x + 2y - 47 = 0x2+y2−2x+2y−47=0

  1. Verification with stored answer

Stored correct answer is D, which matches the derived answer.

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