Circles Questions - JEE

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

The largest value of $\dfrac{y}{x}$, where $(x, y)$ is a real number pair satisfying $(x-3)^2 + (y-3)^2 = 6$, is:

A circle passing through the origin and cutting equal chords of length $\sqrt{2}$ from the straight line $y = x$ and $y = -x$...

A circle is inscribed in an equilateral triangle with side lengths 6 units. Another circle is drawn inside the triangle (but outside the first circle), tangent to the first circle and two of the sides of the triangle. The radius of the smaller circle is:

Consider a series of 'n' concentric circles $C_1, C_2, C_3, ..., C_n$ with radii $r_1, r_2, r_3, ..., r_n$ respectively, such that $n > r_2, ..., > r_n$ and $\sum r_i = 20$. If the tangents drawn from any point on $C_{i+1}$ are such that the chord of contact is a tangent to $C_{i+2}$ $(i = 1, 2, 3, ...)$ and the angle between the tangents from any point on $C_1$ to $C_2$ is $\frac{\pi}{3}$, then find the values of $\lim_{n \to \infty} \sum_{i=1}^{n} r_i$.

Circles Questions (554)