JEE Circles PYQs With Video Solutions, Download PDF

For the four circles $$M, N, O$$ and $$P$$, following four equations are given:
Circle M: $$x^2 + y^2 = 1$$
Circle N: $$x^2 + y^2 - 2x = 0$$
Circle O: $$x^2 + y^2 - 2x - 2y + 1 = 0$$
Circle P: $$x^2 + y^2 - 2y = 0$$
If the centre of circle M is joined with centre of circle N, further centre of circle N is joined with centre of circle O, centre of circle O is joined with the centre of circle P and lastly, centre of circle P is joined with centre of circle M, then these lines form the sides of a:

If the tangent at (1, 7) to the curve $$x^2 = y - 6$$ touches the circle $$x^2 + y^2 + 16x + 12y + c = 0$$ then the value of c is:

The set of all real values of $$\lambda$$ for which exactly two common tangents can be drawn to the circles $$x^2 + y^2 - 4x - 4y + 6 = 0$$ and $$x^2 + y^2 - 10x - 10y + \lambda = 0$$ is the interval:

Let $$r_1$$ and $$r_2$$ be the radii of the largest and smallest circles, respectively, which pass through the point $$(-4, 1)$$ and having their centres on the circumference of the circle $$x^2 + y^2 + 2x + 4y - 4 = 0$$. If $$\frac{r_1}{r_2} = a + b\sqrt{2}$$, then $$a + b$$ is equal to:

The line $$x = y$$ touches a circle at the point (1, 1). If the circle also passes through the point (1, -3), then its radius is

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 ______

Let the straight line $$y = 2x$$ touch a circle with center $$(0, \alpha)$$, $$\alpha \gt 0$$, and radius $$r$$ at a point $$A_1$$. Let $$B_1$$ be the point on the circle such that the line segment $$A_1B_1$$ is a diameter of the circle. Let $$\alpha + r = 5 + \sqrt{5}$$.

Match each entry in List-I to the correct entry in List-II.

Let the point $$P$$ be the vertex of the parabola $$y = x^2 - 6x + 12$$. If a line passing through the point $$P$$ intersects the circle $$x^2 + y^2 - 2x - 4y + 3 = 0$$ at the points $$R$$ and $$S$$.then the maximum value of $$(PR + PS)^2$$ is :

Let 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 $$x - y + 2 = 0$$ intersects the circle at the points $$A$$ and $$B$$, then $$(AB)^2$$ is equal to :

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 the chord $$PQ$$ is $$2$$, then the value of $$h + k + r$$ is equal to ______.

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 five times the distance of point P from chord AB is equal to _____

Let the lengths of intercepts on $$x$$-axis and $$y$$-axis made by the circle $$x^2 + y^2 + ax + 2ay + c = 0$$, $$(a < 0)$$ be $$2\sqrt{2}$$ and $$2\sqrt{5}$$, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line $$x + 2y = 0$$, is equal to:

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 $$(6, 6)$$. If the point $$(6, 6)$$ divides the line segment joining the centres of the circles $$C_1$$ and $$C_2$$ internally in the ratio $$2 : 1$$, then $$(\alpha + \beta) + 4(r_1^2 + r_2^2)$$ equals

Let 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 circle be $$27\sqrt{3}$$. Then the square of the length of the chord of the circle on the line $$x = 1$$ is _______.

If the orthocentre of the triangle formed by the lines $$2x + 3y - 1 = 0$$, $$x + 2y - 1 = 0$$ and $$ax + by - 1 = 0$$, is the centroid of another triangle, whose circumcentre and orthocentre respectively are $$(3, 4)$$ and $$(-6, -8)$$, then the value of $$|a - b|$$ is ________

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 $$3x - 2y = 5$$, then the radius of the circle $$C$$ is

Consider a circle $$(x - \alpha)^2 + (y - \beta)^2 = 50$$, where $$\alpha, \beta > 0$$. If the circle touches the line $$y + x = 0$$ at the point P, whose distance from the origin is $$4\sqrt{2}$$, then $$(\alpha + \beta)^2$$ is equal to _____.

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 $$(2 + c)x + 5c^2y = 1$$. If $$h = \lim_{c \to 1} x_c$$ and $$k = \lim_{c \to 1} y_c$$, then the equation of the circle is :

Let the tangent to the circle $$C_1 : x^2 + y^2 = 2$$ at the point $$M(-1, 1)$$ intersect the circle $$C_2 : (x-3)^2 + (y-2)^2 = 5$$, at two distinct points $$A$$ and $$B$$. If the tangents to $$C_2$$ at the points $$A$$ and $$B$$ intersect at $$N$$, then the area of the triangle $$ANB$$ is equal to

If the area of the triangle formed by the $$x$$-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 equal to ______.

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 $$C_2(\gamma, \delta)$$, $$C_1 \neq C_2$$ are their centres, then $$|(\alpha + \beta)(\gamma + \delta)|$$ is equal to _________.

Two tangents are drawn from a point $$P$$ to the circle $$x^2 + y^2 - 2x - 4y + 4 = 0$$, such that the angle between these tangents is $$\tan^{-1}\left(\frac{12}{5}\right)$$, where $$\tan^{-1}\left(\frac{12}{5}\right) \in (0, \pi)$$. If the centre of the circle is denoted by $$C$$ and these tangents touch the circle at points $$A$$ and $$B$$, then the ratio of the areas of $$\triangle PAB$$ and $$\triangle CAB$$ is:

A circle $$C$$ touches the line $$x = 2y$$ at the point $$(2, 1)$$ and intersects the circle $$C_1 : x^2 + y^2 + 2y - 5 = 0$$ at two points $$P$$ and $$Q$$ such that $$PQ$$ is a diameter of $$C_1$$. Then the diameter of $$C$$ is:

If the circles $$x^2 + y^2 - 16x - 20y + 164 = r^2$$ and $$(x-4)^2 + (y-7)^2 = 36$$ intersect at two distinct points, then:

A circle cuts a chord of length 4a on the x-axis and passes through a point on the y-axis, distant 2b from the origin. Then the locus of the centre of this circle, is:

If $$y + 3x = 0$$ is the equation of a chord of the circle $$x^2 + y^2 - 30x = 0$$, then the equation of the circle with this chord as diameter is:

Let the orthocentre and centroid of a triangle be A(-3, 5) and B(3, 3) respectively. If C is the circumcentre of this triangle, then the radius of the circle having line segment AC as diameter, is:

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 $$S(\alpha, \beta)$$, then $$4\alpha - 7\beta$$ is equal to

Let the maximum and minimum values of $$\left(\sqrt{8x - x^2 - 12} - 4\right)^2 + (x - 7)^2$$, $$x \in \mathbb{R}$$ be $$M$$ and $$m$$, respectively. Then $$M^2 - m^2$$ is equal to _________

Let a circle $$C_1$$ be obtained on rolling the circle $$x^2 + y^2 - 4x - 6y + 11 = 0$$ upwards 4 units on the tangent T to it at the point (3, 2). Let $$C_2$$ be the image of $$C_1$$ in T. Let $$A$$ and $$B$$ be the centers of circles $$C_1$$ and $$C_2$$ respectively, and $$M$$ and $$N$$ be respectively the feet of perpendiculars drawn from $$A$$ and $$B$$ on the x-axis. Then the area of the trapezium AMNB is:

A triangle is formed by the tangents at the point $$(2, 2)$$ on the curves $$y^2 = 2x$$ and $$x^2 + y^2 = 4x$$, and the line $$x + y + 2 = 0$$. If $$r$$ is the radius of its circumcircle, then $$r^2$$ is equal to