Circles Questions - JEE

Two circles have centres at $(a,0)$ and $(-a,0)$ and radii $r_1$ and $r_2$ $(a > r_1 > r_2)$. Then the points of contact of the common tangents to two circles lies on the

A circle $S$ of radius unity touches a line $L$ at $P$. A point $A$ lies on $S$ and $N$ is the foot of the perpendicular from $A$ to $L$. The area of $\triangle PAN$ as $A$ varies cannot be equal to:

Consider the circle $x^2 + y^2 - 10x - 6y + 30 = 0$. Let $O$ be the centre of the circle and tangent at $A(7, 3)$ and $B(5, 1)$ meet at $C$. Let $S = 0$ represents family of circles passing through $A$ and $B$, then:

Let $C_1$ and $C_2$ be centres of two circles whose radii are $2$ and $4$ respectively. Also $C_1C_2 = 10$ and direct common tangents of these circles touch them at $P, Q, R, S$. Another circle of radius $\lambda$ is drawn passing through $P, Q, R, S$. Then

If a circle passes through the point $\left(3, \sqrt{\frac{7}{2}}\right)$ and touches $x + y = 1$ and $x - y = 1$, then the centre of the circle is:

If the conics equations are $S \equiv \sin^2 \theta x^2 + 2h \tan \theta xy + \cos^2 \theta y^2 + 32x + 16y + 19 = 0$, $S' \equiv \cos^2 \theta x^2 + 2h' \cot \theta xy + \sin^2 \theta y^2 + 16x + 32y + 19 = 0$ intersect at four concyclic points, then: (where $\theta \in [0, \pi/2]$)

If largest and smallest value of $\frac{y - 4}{x - 3}$ is $p$ and $q$ where $(x, y)$ satisfy $x^2 + y^2 - 2x - 6y + 9 = 0$ then which of the following is true:

If $A(0,a)$ and $B(0,\beta)$, $a, \beta > 0$ are two vertices of a variable triangle $ABC$, where the vertex $C(x, 0)$ is variable. The value of $x$ for which $\angle ACB$ is maximum is:

Tangents drawn from the point (4, 3) to the circle x2 + y2 - 2x - 4y = 0 are inclined at an angle

If (–3, 2) lies on the circle x² + y² + 2gx + 2fy + c = 0 which is concentric with the circle x² + y² + 6x + 8y - 5 = 0, then c is equal to

Through the point of intersection $P$ of the circle $x^2 + y^2 = 1$ and $x^2 + y^2 = 2x + 4y + 1 = 0$ a common chord $APB$ is drawn terminating on the two circles such that the chords $AP$ and $BP$ of the given circles subtend equal angles at the respective centres. If the coordinates of $P$ are integral and the equation of the chord is $y = 2mx + 1$ then the value of $m$ is ___.

The line lx + my + n = 0 will be a tangent to the circle x2 + y2 = a2 if

Equation of the tangent to the circle, at the point $(1, -1)$, whose centre is the point of intersection of the straight lines $x - y = 1$ and $2x + y = 3$ is

Circles x2 + y2 - 2x - 4y = 0 and x2 + y2 - 8y - 4 = 0

A circle touches the hypotenuse of a right-angled triangle at its middle point and passes through the middle point of the shorter side. If $3$ units and $4$ units be the length of the sides and $'r'$ be the radius of the circle, then find the value of $'3r'$.

Let a circle be given by $2x(x-a) + y(2y-b) = 0$, $a \neq 0$, $b \neq 0$. Find the condition on $a$ and $b$ if two chords, each bisected by the $x$-axis, can be drawn to the circle from the point $\left(a, \dfrac{b}{2}\right)$.

$ABCD$ is rectangle a circle passing through $C$ touches $AB$ and $AD$ at $M$ and $N$ respectively. If the perpendicular distance of $MN$ from $C$ is $5$ then the area of rectangle is ___.

$AB$ is tangent to the circle whose equation is $x^2 + y^2 = 9$. The coordinates of point $A$ are $(-10, 0)$ and point $B(a, b)$ is in the third quadrant. The slope of $AB$ is:

Let the length of common chord be 2a. If two circles satisfy \[\sqrt{9 - a^2} + \sqrt{16 - a^2} = 5\] and the length of common chord is \[\frac{2a}{5} = \frac{k}{5}\], find k.

20. Consider the circle $x^2 + y^2 = 25$ and a point $A(1, 2)$ lying inside it. Next consider secants of the circle passing through point $A$. It turns out that the mid-point of the secants, lie on another circle of centre $(a, b)$ and radius $r$. Then triplet $(a, b, r)$ is:

Six points $(x_i, y_i); i = 1,2,3,4,5,6$ are taken on the circle $x^2 + y^2 = 4$ such that $\sum_{i=1}^{6} x_i = 8$ and $\sum_{i=1}^{6} y_i = 4$. The line segment joining orthocenter of a triangle made by any three points and the centroid of the triangle made by other three points passes through a fixed point $(h, k)$. The value of $h + k$ is ______.

Let x + 2y - 5 + $\lambda$ (x + 3y - 7) = 0 be a variable chord of the circle x2 + y2 - 4x - 6y + 11 = 0. If perpendiculars are drawn at the end points of these chords pass through a fixed point (a, b), a $\neq $ 1 then (a + b) is equal to :

The area of an equilateral triangle inscribed in the circle $x^{2} + y^{2} - 2x = 0$ is

Let the equation of circle be $(x - 3)^2 + (y - 0)^2 + \lambda y = 0$. If it passes through the point $(1, -2)$, find the value of $\lambda$.

If the pair of lines $ax^2 + 2(a+b)xy + by^2 = 0$ lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then

Two circles with centres $C_1$ and $C_2$ have radii $r_1$ and $r_2$ respectively. The circles are such that $C_1C_2 = r_1 + r_2$ and $\sqrt{a^2 + b^2} = 2 \pm \sqrt{a^2 + b^2 - 2}$. If $4r_2 = 2$, find the value of $4r_2$.

From a point $A(2, 2)$ two chords $AB$ and $AC$ of $1$ unit length are drawn to the circle $x^2 + y^2 = 8$. If the equation of the chord $BC$ is given by $ax + by = 15$, then the value of $a + b$ is __________.

If the centre $(\alpha, \beta)$ of a circle lies on the line $y - 4x + 3 = 0$ and the circle passes through the points (2, 3) and (4, 5), find the radius of the circle.

If the equation of tangent to the circle x2 + y2 - 2x + 6y - 6 = 0 and parallel to 3x - 4y + 7 = 0 is 3x - 4y + k = 0, then the value of k are

Let AB be one chord on the circle of centre O at origin (0, 0) by the line $x + y = n$ and D the middle point of the chord. Find the sum of squares of intercepts for $n = 1, 2, 3, 4, 5$ where the circle has equation $x^2 + y^2 = 16$.

The line $y = mx + c$ touches the circle $(x - a)^2 + (y - b)^2 = r^2$ if

The lines $2x - 3y = 5$ and $3x - 4y = 7$ are diameters of a circle having area as 154 sq. units. Then the equation of the circle is

From a variable point P, tangents are drawn one each to the two circles x2 + y2 = a2, x2 + y2 = b2, a > b. If the tangents are mutually perpendicular, then P describes a

Suppose that the equation of the circle having (–3, 5) and (5, –1) as end points of a diameter is $(x – a)^{2} + (y – b)^{2} = r^{2}$. Then a + b + r, (r > 0) is

Four distinct points $(2 k, 3 k),(1,0),(0,1)$ and $(0,0)$ lie on a circle for $k$ equal to:

If the lines $2x + 3y + 1 = 0$ and $3x - y - 4 = 0$ lie along diameters of a circle of circumference $10\pi$, then the equation of the circle is

Three circles touch one another externally. The tangents at their points of contact meet at a point whose distance from point of contact is $4$. Find the ratio of the product of the radii to the double of the sum of the radii of the circles.

The locus of the circumcentre of ∆QAB if q = p/4 is:

The centres of those circles which touch the circle, $x^2 + y^2 - 8x - 8y - 4 = 0$, externally and also touch the $x$-axis, lie on

The angle between a pair of tangents drawn from a point $P$ to the circle $x^2 + y^2 + 4x - 6y + 9\sin^2\alpha + 13\cos^2\alpha = 0$ is $2\alpha$. The equation of the locus of the point $P$ is

The centre of a circle $C$ lies on the line $2x - 2y + 9 = 0$ and this circle cuts $x^2 + y^2 = 4$ orthogonally. If this circle passes through two fixed points $(a, b)$ and $(c, d)$, then the value of $a + b + c + d$ is ___.

In a triangle $ABC$, right angled at $A$, on the leg $AC$ as diameter, a semicircle is described. The chord joining $A$ with the point of intersection $D$ of the hypotenuse and the semicircle, then the length $AC$ equals to:

We have the curves: $x^2 + y^2 = 9$ and $y^2 = 8x$. Let $L_1$ and $L_2$ be the lengths of the common chords of these curves. Then which of the following is true?(Based on the figure and solution: find the ratio $L_1/L_2$.)

Let $BD$ be the internal angle bisector of angle $B$ in triangle $ABC$ with $D$ on side $AC$. The circumcircle of triangle $BDC$ meets $AB$ at $E$, while the circumcircle of triangle $ABD$ meets $BC$ at $F$. If $AE = 3$, then $CF$ is equal to ______.

If ($\alpha$, $\beta$) is the orthocentre of the triangle ABC with vertices A(3, -7), B(-1, 2) and C(4, 5), then 9$\alpha$ - 6$\beta$ + 60 is equal to:

The area of the trapezium ABCD with AB || CD, AD $\perp$ AB and AB = 3CD is equal to 4. A circle inside the trapezium is tangent to all of its sides. If the radius of the circle is r then the value of 4r2, is :

From a figure with points O, P, Q, C, and M, given $OP = 5$, $OQ = 6$, $OM = \frac{5}{2}$, and $CM = 3$. Find $OC^2$ and determine the equation of a circle with centre $\left(\frac{5}{2}, 3\right)$.

A circle passes through the point $(3, 4)$ and cuts the circle $x^2 + y^2 = a^2$ orthogonally. The locus of its centre is a straight line. If the distance of the straight line from the origin is $817$, then find the value of $a^2 - 8140$.

Let $P(a, b)$ be a variable point satisfying $4 \leq a^2 + b^2 \leq 9$ and $b^2 - 4ab + a^2 \leq 0$. Let $R$ be the complete region represented in $x-y$ plane in which $P$ can lie, if $m$ be the minimum value of $|a + b|$ for all position of $P$ lying in region $R$. Then $[m]$ is ___. (Where $[.]$ represents G.I.F.)

In the xy-plane, the length of the shortest path from (0, 0) to (12, 16) that does not go inside the circle (x - 6)^2 + (y - 8)^2 = 25 is

Circles Questions (554)