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Yukti Vault

Coordinate Geometry

Circle: JEE Main & Advanced PYQs

305 previous year questions on Circle from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 51-100 of 305

  1. Let $x^{2}+y^{2}+Ax+By+C=0$ be a circle passing through $(0,6)$ and touching the parabola $y=x^{2}$ at $(2,4)$. Then $A+C$ is equal to :
  2. Let a circle $C$ touch the lines $L_1:4x-3y+K_1=0$ and $L_2:4x-3y+K_2=0$, $K_1,K_2\in\mathbf{R}$. If a line passing through the centre of the circle $C$ intersects $L_1$ …
  3. Let the circles $C_1:(x-\alpha)^{2}+(y-\beta)^{2}=r_1^{2}$ and $C_2:(x-8)^{2}+\left(y-\frac{15}{2}\right)^{2}=r_2^{2}$ touch each other externally at the point $(6,6)$. …
  4. A circle touches both the $y$-axis and the line $x+y=0$. Then the locus of its center is:
  5. Let the locus of the midpoints of the chords of the circle $x^{2}+(y-1)^{2}=1$ drawn from the origin intersect the line $x+y=1$ at $P$ and $Q$. Then, the length of $PQ$ …
  6. If the tangents drawn at the points $O(0,0)$ and $P(1+\sqrt{5},2)$ on the circle $x^{2}+y^{2}-2x-4y=0$ intersect at the point $Q$, then the area of the triangle $OPQ$ is …
  7. Let $C$ be a circle passing through the points $A(2,-1)$ and $B(3,4)$. The line segment $AB$ is not a diameter of $C$. If $r$ is the radius of $C$ and its centre lies on …
  8. Let a circle passing through $(2,0)$ have its centre at the point $(h,k)$. Let $(x_c,y_c)$ be the point of intersection of the lines $3x+5y=1$ and $(2+c)x+5c^{2}y=1$. If …
  9. Let $C_1$ be the circle in the third quadrant of radius $3$, that touches both coordinate axes. Let $C_2$ be the circle with centre $(1,3)$ that touches $C_1$ externally …
  10. If one of the diameters of the circle $x^{2}+y^{2}-10x+4y+13=0$ is a chord of another circle $C$, whose center is the point of intersection of the lines $2x+3y=12$ and …
  11. If the image of the point $(-4,5)$ in the line $x+2y=2$ lies on the circle $(x+4)^{2}+(y-3)^{2}=r^{2}$, then $r$ is equal to :
  12. If the circle $x^2+y^2-2gx+6y-19c=0$, $g,c\in\mathbb{R}$ passes through the point $(6,1)$ and its centre lies on the line $x-2cy=8$, then the length of intercept made by …
  13. The circle passing through the intersection of the circles, $x^2+y^2-6x=0$ and $x^2+y^2-4y=0$, having its centre on the line, $2x-3y+12=0$, also passes through the point …
  14. The circles, $x^{2}+y^{2}+2x+4y-4=0$ and $x^{2}+y^{2}-4x+4y+k=0$ touch each other internally. If the point of contact of these circles is $(a,b)$, then …
  15. A circle passing through the point $P(\alpha,\beta)$ in the first quadrant touches the two coordinate axes at the points $A$ and $B$. The point $P$ is above the line …
  16. Equations of two diameters of a circle are $2x-3y=5$ and $3x-4y=7$. The line joining the points $\left(-\frac{22}{7},-4\right)$ and $\left(-\frac{1}{7},3\right)$ …
  17. A tangent line $L$ is drawn at the point $(2,-4)$ on the parabola $y^{2}=8x$. If the line $L$ is also tangent to the circle $x^{2}+y^{2}=a$, then '$a$' is equal to …
  18. The locus of a point, which moves such that the sum of squares of its distances from the points $(0,0)$, $(1,0)$, $(0,1)$, $(1,1)$ is $18$ units, is a circle of diameter …
  19. The number of integral values of $k$ for which the line, $3x+4y=k$ intersects the circle, $x^2+y^2-2x-4y+4=0$ at two distinct points is ________.
  20. Let $AB$ be a chord of length $12$ of the circle $(x-2)^2+(y+1)^2=\frac{169}{4}$. If tangents drawn to the circle at points $A$ and $B$ intersect at the point $P$, then …
  21. If the curves, $x^2-6x+y^2+8=0$ and $x^2-8y+y^2+16-k=0$, ($k>0$) touch each other at a point, then the largest value of $k$ is ________.
  22. Two circles each of radius $5$ units touch each other at the point $(1,2)$. If the equation of their common tangent is $4x+3y=10$, and $C_1(\alpha,\beta)$ and …
  23. Let the equation $x^{2}+y^{2}+px+(1-p)y+5=0$ represent circles of varying radius $r\in(0,5]$. Then the number of elements in the set $S=\{q:q=p^{2}$ and $q$ is an …
  24. Let the point $(p,p+1)$ lie inside the region $E=\left\{(x,y):3-x\le y\le\sqrt{9-x^{2}},\ 0\le x\le3\right\}$. If the set of all values of $p$ is the interval $(a,b)$ …
  25. If the variable line $3x+4y=\alpha$ lies between the two circles $(x-1)^{2}+(y-1)^{2}=1$ and $(x-9)^{2}+(y-1)^{2}=4$, without intercepting a chord on either circle, then …
  26. 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 …
  27. Let $B$ be the centre of the circle $x^{2}+y^{2}-2x+4y+1=0$. Let the tangents at two points $P$ and $Q$ on the circle intersect at the point $A(3,1)$. Then …
  28. If the circles $x^2+y^2+6x+8y+16=0$ and $x^2+y^2+2(3-\sqrt3)x+2(4-\sqrt6)y=k+6\sqrt3+8\sqrt6$, $k>0$, touch internally at the point $P(\alpha,\beta)$, then …
  29. Let the mirror image of a circle $c_1:x^2+y^2-2x-6y+\alpha=0$ in line $y=x+1$ be $c_2:5x^2+5y^2+10gx+10fy+38=0$. If $r$ is the radius of circle $c_2$, then $\alpha+6r^2$ …
  30. Let a circle $C:(x-h)^{2}+(y-k)^{2}=r^{2}$, $k>0$, touch the $x$-axis at $(1,0)$. If the line $x+y=0$ intersects the circle $C$ at $P$ and $Q$ such that the length of …
  31. Let a point $P$ be such that its distance from the point $(5,0)$ is thrice the distance of $P$ from the point $(-5,0)$. If the locus of the point $P$ is a circle of …
  32. The diameter of the circle, whose centre lies on the line $x+y=2$ in the first quadrant and which touches both the lines $x=3$ and $y=2$, is ________.
  33. Let a circle $C$ of radius $5$ lie below the $x$-axis. The line $L_1:4x+3y+2=0$ passes through the centre $P$ of the circle $C$ and intersects the line $L_2:3x-4y-11=0$ …
  34. Let the centre of a circle, passing through the points $(0,0)$, $(1,0)$ and touching the circle $x^{2}+y^{2}=9$, be $(h,k)$. Then for all possible values of the …
  35. If one of the diameters of the circle $x^{2}+y^{2}-2\sqrt{2}x-6\sqrt{2}y+14=0$ is a chord of the circle …
  36. 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 …
  37. Two circles in the first quadrant of radii $r_1$ and $r_2$ touch the coordinate axes. Each of them cuts off an intercept of $2$ units with the line $x+y=2$. Then …
  38. Let the lines $y+2x=\sqrt{11}+7\sqrt{7}$ and $2y+x=2\sqrt{11}+6\sqrt{7}$ be normal to a circle $C:(x-h)^{2}+(y-k)^{2}=r^{2}$. If the line …
  39. The minimum distance between any two points $P_1$ and $P_2$ while considering point $P_1$ on one circle and point $P_2$ on the other circle for the given circles' …
  40. Consider a circle $C_1:x^{2}+y^{2}-4x-2y=\alpha-5$. Let its mirror image in the line $y=2x+1$ be another circle $C_2:5x^{2}+5y^{2}-10fx-10gy+36=0$. Let $r$ be the radius …
  41. Let $PQ$ be a diameter of the circle $x^2+y^2=9$. If $\alpha$ and $\beta$ are the lengths of the perpendiculars from $P$ and $Q$ on the straight line, $x+y=2$ …
  42. Let the abscissae of the two points $P$ and $Q$ be the roots of $2x^{2}-rx+p=0$ and the ordinates of $P$ and $Q$ be the roots of $x^{2}-sx-q=0$. If the equation of the …
  43. If the area of the triangle formed by the positive $x$-axis, the normal and the tangent to the circle $(x-2)^{2}+(y-3)^{2}=25$ at the point $(5,7)$ is $A$, then $24A$ is …
  44. A line is a common tangent to the circle $(x-3)^{2}+y^{2}=9$ and the parabola $y^{2}=4x$. If the two points of contact $(a,b)$ and $(c,d)$ are distinct and lie in the …
  45. Consider two circles $C_1:x^{2}+y^{2}=25$ and $C_2:(x-\alpha)^{2}+y^{2}=16$, where $\alpha\in(5,9)$. Let the angle between the two radii (one to each circle) drawn from …
  46. A circle with centre $(2,3)$ and radius $4$ intersects the line $x+y=3$ at the points $P$ and $Q$. If the tangents at $P$ and $Q$ intersect at the point …
  47. Let $ABC$ be the triangle with $AB=1$, $AC=3$ and $\angle BAC=\frac{\pi}{2}$. If a circle of radius $r>0$ touches the sides $AB$, $AC$ and also touches internally the …
  48. A line $y=mx+1$ intersects the circle $(x-3)^2+(y+2)^2=25$ at the points $P$ and $Q$. If the midpoint of the line segment $PQ$ has $x$-coordinate $-\frac{3}{5}$, then …
  49. Find the equations of the circles passing through $(-4,3)$ and touching the lines $x+y=2$ and $x-y=2$.
  50. Let $P(a_1,b_1)$ and $Q(a_2,b_2)$ be two distinct points on a circle with center $C\left(\sqrt2,\sqrt3\right)$. Let $O$ be the origin and $OC$ be perpendicular to both …

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