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

2022 · 27 Jul · Shift 1 · Q37
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. /2022 · 27 Jul · Shift 1 · Q37

Circle question

2022 · 27 Jul · Shift 1 · Q37

JEE MainMathematicsCircleMCQ+4 / −1
If the circle x2+y2−2gx+6y−19c=0,g,c∈Rx^{2}+y^{2}-2 g x+6 y-19 c=0, g, c \in \mathbb{R}x2+y2−2gx+6y−19c=0,g,c∈R passes through the point (6,1)(6,1)(6,1) and its centre lies on the line x−2cy=8x-2 c y=8x−2cy=8, then the length of intercept made by the circle on xxx-axis is :
  1. A
    11\sqrt{11}11​
  2. B
    4
  3. C
    3
  4. D
    2232 \sqrt{23}223​
View written solutionFree

Correct answer: D

  1. Write the circle in standard form

    Given: x2+y2−2gx+6y−19c=0x^2+y^2-2gx+6y-19c=0x2+y2−2gx+6y−19c=0

    Compare with the general form: x2+y2+2ux+2vy+w=0x^2+y^2+2ux+2vy+w=0x2+y2+2ux+2vy+w=0 whose centre is (−u,−v)(-u,-v)(−u,−v).

    Here,

    \qquad 6=2v \Rightarrow v=3$$ so the centre is $$(-u,-v)=(g,-3).$$

  2. Use the condition that the centre lies on x−2cy=8x-2cy=8x−2cy=8

    Since the centre is (g,−3)(g,-3)(g,−3), substitute into the line: g−2c(−3)=8g-2c(-3)=8g−2c(−3)=8 g+6c=8g+6c=8g+6c=8 g=8−6c.g=8-6c. g=8−6c.

  3. Use the condition that the circle passes through (6,1)(6,1)(6,1)

    Substitute (x,y)=(6,1)(x,y)=(6,1)(x,y)=(6,1) into the circle: 62+12−2g(6)+6(1)−19c=06^2+1^2-2g(6)+6(1)-19c=062+12−2g(6)+6(1)−19c=0 36+1−12g+6−19c=036+1-12g+6-19c=036+1−12g+6−19c=0 43−12g−19c=0.43-12g-19c=0. 43−12g−19c=0.

    Now substitute g=8−6cg=8-6cg=8−6c: 43−12(8−6c)−19c=043-12(8-6c)-19c=043−12(8−6c)−19c=0 43−96+72c−19c=043-96+72c-19c=043−96+72c−19c=0 53c−53=053c-53=053c−53=0 c=1.c=1. c=1.

    Hence, g=8−6(1)=2.g=8-6(1)=2. g=8−6(1)=2.

  4. Write the actual circle

    Substituting g=2,c=1g=2, c=1g=2,c=1: x2+y2−4x+6y−19=0.x^2+y^2-4x+6y-19=0. x2+y2−4x+6y−19=0.

  5. Find the intercept on the xxx-axis

    On the xxx-axis, y=0y=0y=0. So: x2−4x−19=0.x^2-4x-19=0. x2−4x−19=0.

    Solve:

    =\frac{4\pm\sqrt{92}}{2} =\frac{4\pm 2\sqrt{23}}{2} =2\pm\sqrt{23}. $$ Thus the two intersection points with the $x$-axis are: $$(2-\sqrt{23},0), \quad (2+\sqrt{23},0).$$ Therefore, the length of the intercept is: $$[(2+\sqrt{23})-(2-\sqrt{23})]=2\sqrt{23}. $$

  6. Check options

    2232\sqrt{23}223​ matches Option D.

PreviousNext

More from Circle

  • A rectangle R with end points of one of its sides as (1, 2) and (3, 6) is inscribed in a circle. If the equation of a diameter of the circle is 2x − y + 4 = 0, then the area of R is ​.2022 · Numerical
  • The set of values of k, for which the circle C:4x2+4y2−12x+8y+k=0 lies inside the fourth quadrant and the point (1,−31​) lies on or inside the circle C, is :2022 · MCQ
  • For t∈(0,2π), if ABC is an equilateral triangle with vertices A(sint,−cost),B(cost,sint) and C(a,b) such that its orthocentre lies on a circle with centre…2022 · MCQ
  • Let C be the centre of the circle x2+y2−x+2y=411​ and P be a point on the circle. A line passes through the point C, makes an angle of 4π​ with the line CP and intersects the circle at…2022 · MCQ
  • If one of the diameters of the circle x2+y2−22​x−62​y+14=0 is a chord of the circle (x−22​)2+(y−22​)2=r2, then the value of r2 is equal to ​.2022 · Numerical
  • Let the mirror image of a circle c1​:x2+y2−2x−6y+α=0 in line y=x+1 be c2​:5x2+5y2+10gx+10fy+38=0. If r is the radius of circle c2​, then α+6r2 is equal to ​…2022 · Numerical
  • Let AB be a chord of length 12 of the circle (x−2)2+(y+1)2=4169​. If tangents drawn to the circle at points A and B intersect at the point P, then five times the distance of point P from chord AB is equal to ​…2022 · Numerical
  •  Let S={(x,y)∈N×N:9(x−3)2+16(y−4)2≤144} and T={(x,y)∈R×R:(x−7)2+(y−4)2≤36}. Then n(S∩T) is equal to ​…2022 · Numerical