Sign in
12thPass logo
New chatPYQ LibraryDoubtsRank report
Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →
Sign in

Circle question

2004 · Shift 0 · Q111
Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.
  1. PYQ Library
  2. /JEE Main
  3. /Mathematics
  4. /Circle
  5. /2004 · Shift 0 · Q111

Circle question

2004 · Shift 0 · Q111

JEE MainMathematicsCircleMCQ+4 / −1
Intercept on the line y = x by the circle x2 + y2−2x=0{x^2}\, + \,{y^2} - 2x = 0x2+y2−2x=0 is AB. Equation of the circle on AB as a diameter is :
  1. A
     x2 + y2+ x − y  =0\,{x^2}\, + \,{y^2} + \,x\, - \,y\,\, = 0x2+y2+x−y=0
  2. B
     x2 + y2− x + y  =0\,{x^2}\, + \,{y^2} - \,x\, + \,y\,\, = 0x2+y2−x+y=0
  3. C
     x2 + y2+ x + y  =0\,{x^2}\, + \,{y^2} + \,x\, + \,y\,\, = 0x2+y2+x+y=0
  4. D
     x2 + y2− x − y  =0\,{x^2}\, + \,{y^2} - \,x\, - \,y\,\, = 0x2+y2−x−y=0
View written solutionFree

Correct answer: D

  1. Given circle

    The circle is x2+y2−2x=0.x^2+y^2-2x=0.x2+y2−2x=0.

    Rewrite it in standard form: x2−2x+y2=0x^2-2x+y^2=0x2−2x+y2=0 x2−2x+1+y2=1x^2-2x+1+y^2=1x2−2x+1+y2=1 (x−1)2+y2=1.(x-1)^2+y^2=1.(x−1)2+y2=1.

    So its center is (1,0)(1,0)(1,0) and radius is 111.

  2. Find the points of intersection with the line y=xy=xy=x

    Substitute y=xy=xy=x into the circle equation: x2+x2−2x=0x^2+x^2-2x=0x2+x2−2x=0 2x2−2x=02x^2-2x=02x2−2x=0 2x(x−1)=0.2x(x-1)=0.2x(x−1)=0.

    Hence, x=0orx=1.x=0 \quad \text{or} \quad x=1.x=0orx=1.

    Since y=xy=xy=x, the intersection points are: A=(0,0),B=(1,1).A=(0,0), \qquad B=(1,1).A=(0,0),B=(1,1).

  3. Equation of circle with ABABAB as diameter

    If (x1,y1)(x_1,y_1)(x1​,y1​) and (x2,y2)(x_2,y_2)(x2​,y2​) are endpoints of a diameter, then the circle is given by (x−x1)(x−x2)+(y−y1)(y−y2)=0.(x-x_1)(x-x_2) + (y-y_1)(y-y_2)=0.(x−x1​)(x−x2​)+(y−y1​)(y−y2​)=0.

    Using A=(0,0)A=(0,0)A=(0,0) and B=(1,1)B=(1,1)B=(1,1): x(x−1)+y(y−1)=0.x(x-1)+y(y-1)=0.x(x−1)+y(y−1)=0.

    Expanding: x2−x+y2−y=0x^2-x+y^2-y=0x2−x+y2−y=0 x2+y2−x−y=0.x^2+y^2-x-y=0.x2+y2−x−y=0.

  4. Match with options

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

  5. Comparison with stored answer

    Stored correct answer: D

    Our derived answer: D

    So they agree.

PreviousNext

More from Circle

  • The lines 2x - 3y = 5 and 3x - 4y = 7 are diameters of a circle having area as 154 sq. units. Then the equation of the circle is :2003 · MCQ
  • If the chord y = mx + 1 of the circle x2+y2=1 subtends an angle of measure 45∘ at the major segment of the circle then value of m is :2002 · MCQ
  • The equation of a circle with origin as a center and passing through an equilateral triangle whose median is of length 3a is :2002 · MCQ
  • The absolute difference between the squares of the radii of the two circles passing through the point (−9,4) and touching the lines x+y=3 and x−y=3, is equal to ​ .2025 · Numerical
  • If the four distinct points (4,6),(−1,5),(0,0) and (k,3k) lie on a circle of radius r, then 10k+r2 is equal to2025 · MCQ
  • Let C be the circle x2+(y−1)2=2,E1​ and E2​ be two ellipses whose centres lie at the origin and major axes lie on x -axis and y -axis respectively. Let the straight line x+y=3 touch the curves C,E1​ and E2​ at P(x1​,y1​),Q(x2​,y2​)…2025 · Numerical
  • Let C1​ be the circle in the third quadrant of radius 3 , that touches both coordinate axes. Let C2​ be the circle with centre (1,3) that touches C1​ externally at the point (α,β). If (β−α)2=nm​…2025 · MCQ
  • A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at the point (2,5) and intersects the circle C at exactly two points. If the set of all possible…2025 · MCQ