Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE MainMathematicsCircle

If the circles x^2 + y^2 = 49 and x^2 + y^2 - 2kx - 6y + k^2 + 5 = 0 intersect at exactly two distinct points, then the range of values of k is

Options

  1. Ak (-6 2 , 6 2 )
  2. Bk (- , -4) (4, )
  3. Ck (-4, 4)
  4. Dk (-6 2 , -4) (4, 6 2 )

Correct answer

D. k (-6 2 , -4) (4, 6 2 )

Step-by-step solution

Let the given circles be S₁ x^2 + y^2 = 49 and S₂ x^2 + y^2 - 2kx - 6y + k^2 + 5 = 0 . The center and radius of S₁ are C₁(0, 0) and r₁ = 7 . The center and radius of S₂ are C₂(k, 3) and r₂ = k^2 + 9 - (k^2 + 5) = 4 = 2 . For two circles to intersect at exactly two distinct points, the distance between their centers d must satisfy: |r₁ - r₂| The distance between the centers is d = k^2 + 3^2 = k^2 + 9 . Substituting the values into the condition: |7 - 2| 5 Squaring all parts of the inequality: 25 16 This gives two in

Practice Circle on Quantrex Academy →

More from Circle

Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I List-II (P) The circle with centre (1, 2) and touching the straight line 3x + 4y = 2026Consider the circle C: x^2+y^2-6x-8y-11=0 . Let a variable chord AB of the circle C subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from th 2026Let C be a circle having centre in the first quadrant and touching the x -axis at a distance of 3 units from the origin. If the circle C has an intercept of length 6 3 on y -axis, 2026Let the line x - y = 4 intersect the circle C: (x-4)^2 + (y+3)^2 = 9 at the points Q and R . If P( , ) is a point on C such that PQ = PR , then (6 + 8 )^2 is equal to __________. 2026Let the centre of the circle x^2 + y^2 + 2gx + 2fy + 25 = 0 be in the first quadrant and lie on the line 2x - y = 4 . Let the area of an equilateral triangle inscribed in the circl 2026Let P be a moving point on the circle x^2 + y^2 - 6x - 8y + 21 = 0 . Then, the maximum distance of P from the vertex of the parabola x^2 + 6x + y + 13 = 0 is equal to: 2026Suppose that two chords, drawn from the point (1, 2) on the circle x^2 + y^2 + x - 3y = 0 are bisected by the y -axis. If the other ends of these chords are R and S , and the mid p 2026Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines x + (k-1)y + 3 = 0 and 2x + k^2 y - 4 = 0 . If the line x - y + 2026 Full Circle list All JEE Main PYQs