Circles-JEE-Main-PYQ’s

Mathematics Questions

Let the circle $x^2 + y^2 = 4$ intersect $x$ -axis at the points $A(a, 0), a > 0$ and $B(b, 0)$ . Let $P(2 \cos \alpha, 2 \sin \alpha), 0 < \alpha < \frac{\pi}{2}$ and $Q(2 \cos \beta, 2 \sin \beta)$ be two points such that $(\alpha - \beta) = \frac{\pi}{2}$ . Then the point of intersection of $AQ$ and $BP$ lies on :

(A) $x^2 + y^2 - 4x - 4 = 0$

(B) $x^2 + y^2 - 4x - 4y = 0$

(D) $x^2 + y^2 - 4y - 4 = 0$

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Let $y = x$ be the equation of a chord of the circle $C_1$ (in the closed half-plane $x \geq 0$ ) of diameter 10 passing through the origin. Let $C_2$ be another circle described on the given chord as its diameter. If the equation of the chord of the circle $C_2$ , which passes through the point $(2, 3)$ and is farthest from the center of $C_2$ , is $x + ay + b = 0$ , then $a - b$ is equal to

(A) -6

(B) 10

(D) -2

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Let a circle of radius 4 pass through the origin O, the points $A(-\sqrt{3}a, 0)$ and $B(0, -\sqrt{2}b)$ , where $a$ and $b$ are real parameters and $ab \neq 0$ . Then the locus of the centroid of $\triangle OAB$ is a circle of radius

(A) $\frac{7}{3}$

(B) $\frac{11}{3}$

(D) $\frac{8}{3}$

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Let the set of all values of $r$ , for which the circles $(x + 1)^2 + (y + 4)^2 = r^2$ and $x^2 + y^2 - 4x - 2y - 4 = 0$ intersect at two distinct points be the interval $(\alpha, \beta)$ . Then $\alpha\beta$ is equal to

(A) 21

(B) 24

(D) 25

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Let $PQ$ and $MN$ be two straight lines touching the circle $x^2 + y^2 - 4x - 6y - 3 = 0$ at the points $A$ and $B$ respectively. Let $O$ be the centre of the circle and $\angle AOB = \pi/3$ . Then the locus of the point of intersection of the lines $PQ$ and $MN$ is :

(A) $x^2 + y^2 - 18x - 12y - 25 = 0$

(B) $x^2 + y^2 - 12x - 18y - 25 = 0$

(D) $3(x^2 + y^2) - 18x - 12y + 25 = 0$

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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 values of $r$ is the interval $(\alpha, \beta)$ , then $3\beta - 2\alpha$ is equal to:

(A) $10$

(B) $12$

(D) $15$

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Let circle $C$ be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ in the line $2x - 3y + 5 = 0$ and $A$ be the point on $C$ such that $OA$ is parallel to $x$ -axis and $A$ lies on the right hand side of the centre $O$ of $C$ . If $B(\alpha, \beta)$ , with $\beta < 4$ , lies on $C$ such that the length of the arc $AB$ is $(1/6)^{\text{th}}$ of the perimeter of $C$ , then $\beta - \sqrt{3}\alpha$ is equal to:

(A) $4 - \sqrt{3}$

(B) $3$

(D) $3 + \sqrt{3}$

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Let the equation of the circle, which touches $x$ -axis at the point $(a, 0)$ , $a > 0$ and cuts off an intercept of length $b$ on $y$ -axis be $x^2 + y^2 - \alpha x + \beta y + \gamma = 0$ . If the circle lies below $x$ -axis, then the ordered pair $(2a, b^2)$ is equal to:

(A) $(\alpha, \beta^2 + 4\gamma)$

(B) $(\alpha, \beta^2 - 4\gamma)$

(D) $(\gamma, \beta^2 + 4\alpha)$

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Let the line $x + y - 1 = 0$ meet the circle $x^2 + y^2 = 4$ at the points $A$ and $B$ . If the line perpendicular to $AB$ and passing through the mid-point of the chord $AB$ intersects the circle at $C$ and $D$ , then the area of the quadrilateral $ABCD$ is equal to:

(A) $\sqrt{14}$

(B) $3\sqrt{7}$

(D) $5\sqrt{7}$

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Let a circle $C$ pass through the points $(4, 2)$ and $(0, 2)$ , and its centre lie on $3x + 2y + 2 = 0$ . Then the length of the chord of the circle $C$ , whose mid-point is $(1, 2)$ , is: