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Circle question

2004 · Shift 0 · Q109
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Circle question

2004 · Shift 0 · Q109

JEE MainMathematicsCircleMCQ+4 / −1
If the lines 2x + 3y + 1 + 0 and 3x - y - 4 = 0 lie along diameter of a circle of circumference 10 π10\,\pi10π, then the equation of the circle is :
  1. A
    x2 + y2+ 2x − 2y− 23  =0{x^2}\, + \,{y^2} + \,2x\, - \,2y - \,23\,\, = 0x2+y2+2x−2y−23=0
  2. B
    x2 + y2− 2x − 2y− 23  =0{x^2}\, + \,{y^2} - \,2x\, - \,2y - \,23\,\, = 0x2+y2−2x−2y−23=0
  3. C
    x2 + y2+ 2x + 2y− 23  =0{x^2}\, + \,{y^2} + \,2x\, + \,2y - \,23\,\, = 0x2+y2+2x+2y−23=0
  4. D
    x2 + y2− 2x + 2y− 23  =0{x^2}\, + \,{y^2} - \,2x\, + \,2y - \,23\,\, = 0x2+y2−2x+2y−23=0
View written solutionFree

Correct answer: D

  1. Interpret the statement

The two lines 2x+3y+1=02x+3y+1=02x+3y+1=0 and 3x−y−4=03x-y-4=03x−y−4=0 intersect at a point. Since they "lie along diameter" of the circle, their point of intersection is the center of the circle.

Also, the circumference of the circle is given as 10π.10\pi.10π.

Using 2πr=10π,2\pi r=10\pi,2πr=10π, we get r=5.r=5.r=5.

  1. Find the center by solving the two lines

We solve: 2x+3y+1=0⇒2x+3y=−12x+3y+1=0 \quad \Rightarrow \quad 2x+3y=-12x+3y+1=0⇒2x+3y=−1 3x−y−4=0⇒3x−y=43x-y-4=0 \quad \Rightarrow \quad 3x-y=43x−y−4=0⇒3x−y=4

From the second equation, y=3x−4.y=3x-4.y=3x−4.

Substitute into the first: 2x+3(3x−4)=−12x+3(3x-4)=-12x+3(3x−4)=−1 2x+9x−12=−12x+9x-12=-12x+9x−12=−1 11x=1111x=1111x=11 x=1.x=1.x=1.

Then y=3(1)−4=−1.y=3(1)-4=-1.y=3(1)−4=−1.

So the center is (1,−1).(1,-1).(1,−1).

  1. Form 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.

Here, h=1,k=−1,r=5.h=1,\quad k=-1,\quad r=5.h=1,k=−1,r=5.

So, (x−1)2+(y+1)2=25.(x-1)^2+(y+1)^2=25.(x−1)2+(y+1)2=25.

Expand: x2−2x+1+y2+2y+1=25x^2-2x+1+y^2+2y+1=25x2−2x+1+y2+2y+1=25 x2+y2−2x+2y−23=0.x^2+y^2-2x+2y-23=0.x2+y2−2x+2y−23=0.

  1. Match with the options

This is x2+y2−2x+2y−23=0,x^2+y^2-2x+2y-23=0,x2+y2−2x+2y−23=0, which matches Option D.

  1. Comparison with stored answer

Stored correct answer: D

Our derived answer: D

So the answer agrees with the stored answer.

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