Circles Questions - JEE

A circle $C$ passes through the points of intersection of the parabola $y + 1 = (x - 4)^2$ and the $x$-axis. The length of tangent from origin to $C$ is:

Let the bisector of ∠A of △ABC meets BC in D and the circumcircle of △ABC in E. Analyze: AD is less than the G.M. (Geometric Mean) of AB and AC.

If S₁, S₂ and S₃ are three circles congruent to C₂ and touch both C₁ and C₂; then the area of triangle formed by joining centres of the circles S₁, S₂ and S₃ is (in square units)

Let A = (0, 0), B = (4, 0) and on segment AB is given a point M. On the same side of AB, squares AMCD and BMFE are constructed above AB. The circumcircles S₁ and S₂ of two squares AMCD and BMFE respectively have centres P and Q, and intersect in M and another point N.For all positions of M varying along the segment AB, the line MN passes through the fixed point R(a, b), then a + b = ?

Locus of the image of the point (2, 3) in the line $(2x - 3y + 4) + k(x - 2y + 3) = 0,\ k \in \mathbb{R}$, is a

Let B and C be two fixed points of a given circle and A a variable point lying on major arc of this circle. The locus of feet of perpendiculars dropped from the midpoint of AB on AC is:

Given a line segment AB, where A = (0,0) and B = (a,0). Three circles S₁, S₂, S₃ of radius R are centred at the end points and the midpoint of the line segment AB. A fourth circle S₄ is drawn touching the 3 given circles.If \(\frac{a}{4}

Given a line segment AB, where A is at (0, 0) and B at (a, 0). Three circles S₁, S₂, S₃ of radius R are centred at the endpoints and the midpoint of the line segment AB. A fourth circle S₄ is drawn touching the 3 given circles.If \(\frac{a}{4}

Given a line segment AB, A ≡ (0, 0) and B(a, 0). Three circles S₁, S₂, S₃ of radius R are centred at the end points and the midpoint of the line segment AB. A fourth circle S₄ is drawn touching the 3 given circles. If a/4

In a right 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:

Let C be the circle of radius unity centred at the origin. If two positive numbers $x_1$ and $x_2$ are such that the line passing through $(x_1, -1)$ and $(x_2, 1)$ is tangent to C, then:

P is a point $(a, b)$ in the first quadrant. If the two circles which pass through P and touch both the coordinate axes cut at right angles, then:

Let ABC be an equilateral triangle inscribed in C. If a, b, γ denote the distances of D from vertices A, B, C respectively, what is the value of the product $\frac{(b + \gamma - a)(\gamma + a - b)(a + b - \gamma)}{abg}$:

Let C be a circle with centre O and HK is the chord of contact of tangents drawn from a point A. OA intersects the circle C at P and Q, and B is the midpoint of HK. Prove or disprove: AB is the Harmonic Mean of AP and AQ.

Let $S=\left\{z\in\mathbb{C}:\left|\dfrac{z-6i}{z-2i}\right|=1\text{ and }\left|\dfrac{z-8+2i}{z+2i}\right|=\dfrac{3}{5}\right\}$. Then $\displaystyle\sum_{z\in S}|z|^2$ is equal to

Let a circle be inscribed in the quadrant of a circle of diameter 4.Statement-1: The radius of inscribed circle is the positive root of the equation $r^2 + 4r - 4 = 0$Statement-2: Distance between their centres = $2$ (radius of circle inscribed).

If 2x - 4y = 9 and 6x -12y + 7 = 0 are the tangents of same circle, then its radius will be

If two parallel chords of a circle, having diameter 4 units, lie on the opposite sides of the centre and subtend angles $\cos^{-1}(1/7)$ and $\sec^{-1}(7)$ at the centre, respectively, then the distance between these chords is

$A$, $B$ and $C$ are points in the xy-plane such that $A(1, 2)$; $B(5, 6)$ and $AC = 3BC$. Then:

In a Δ ABC; if (II₁)² + (I₂I₃)² = lR², where I denotes incentre; I₁, I₂ and I₃ denote centres of the circles escribed to the sides BC, CA and AB respectively and R be the radius of the circumcircle of Δ ABC. Find l.

Let PQ be fixed chord in a circle S and AB is any diameter.Statement-1: If PQ is represented by equation $y = -1$, S represented by $x^2 + y^2 = 4$ and A, B lie on same side of PQ then the sum of the perpendiculars let fall from A and B on PQ is equal to 4.Statement-2: The sum of the perpendiculars let fall from A and B on PQ is same for all position of AB.

If the radius of the circle $(x-1)^2 + (y-2)^2 = 1$ and $(x-7)^2 + (y-10)^2 = 4$ are increasing uniformly w.r.t. times as 0.3 and 0.4 unit/sec, then they will touch each other at $t$ equal to

Let A = (0, 0), B = (4, 0) and on segment AB is given a point M. On the same side of AB, squares AMCD and BMFE are constructed above AB. The circumcircles S₁ and S₂ of two squares AMCD and BMFE respectively have centres P and Q, and intersect in M and another point N.The point of intersection of the lines FA and BC is:

An altitude BD and a bisector BE are drawn in the triangle ABC from the vertex B. It is known that the length of side AC = 1, and the magnitudes of the angles $\angle BEC$, $\angle ABD$, $\angle ABE$, $\angle BAC$ form an arithmetic progression.The area of circle circumscribing $\triangle ABC$ is:

The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is $60°$. If the area of the quadrilateral is $4\sqrt{3}$, then the perimeter of the quadrilateral is

If two equal chords AB and CD of a circle intersect at point P, where ACB and CBD are minor segments of AB and CD, AP = 2 and PB = 7, then prove or disprove: PC = 7 and PD = 2.

For any real k, the circle $x^2 + y^2 + 2kx + 2ky - 8 = 0$ passes through two fixed points A and B. Locus of the point of intersection of the tangents to the circle at A and B is:

Let 'O' be the circumcentre of $\triangle ABC$. Under the same conditions as Problem 1, the radius of circle inscribed in $\triangle BOC$ is:

PA and PB are two tangents drawn from point P to circle of radius 5. A line is drawn from point P which cuts circle at C and D such that PC = 5 and PD = 15 and $\angle APB = \theta$. Find the area of $\triangle APB$.

Let ABCD be a rectangle with diagonals AC and BD intersecting at O. A straight line through B intersects DC produced at E and DA produced at F such that OE = OF. Prove or disprove: $(CE)(DE) = (AF)(DF)$.

Given $x^2 + y^2 - 16x - 20y + 164 = r^2$. If the circles intersect at two points, then the range of $r$ is:

Let ABC be an equilateral triangle inscribed in C. If a, b, γ denote the distances of D from vertices A, B, C respectively, what is the value of the product $\frac{(b + \gamma - a)(\gamma + a - b)(a + b - \gamma)}{abg}$:

Let A = (0,0), B = (4,0) and on segment AB is given a point M. On the same side of AB, squares AMCD and BMFE are constructed above AB. The circumcircles S₁ and S₂ of two squares AMCD and BMFE respectively have centres P and Q, and intersect in M and another point N.For all positions of M varying along the segment AB, the line MN passes through the fixed point R(a,b). Then a + b =

The circle S touches the sides AB and AD of the rectangle ABCD and cuts the side DC at a single point F and the side BC at a single point E. If $|AB| = 32$, $|AD| = 40$ and $|BE| = 1$. The radius of circle S is:

In the diagram, DC is a diameter of the large circle centered at A, and AC is a diameter of the smaller circle centered at B. If DE is tangent to the smaller circle at F and DC = 12, then the length of DE is:

The locus of the point of intersection of the tangent to the circle $x^2 + y^2 = a^2$, which include an angle of $45°$ is the curve $(x^2 + y^2)^2 = \lambda a^2(x^2 + y^2 - a^2)$. The value of $\lambda$ is:

In an acute triangle ABC, point H is the intersection point of altitude CE to AB and altitude BD to AC. A circle with DE as its diameter intersects AB and AC at points F and G respectively. If BC = 25, BD = 20 and BE = 7. Area of the circle S is:

Suppose that two circles $C_1$ and $C_2$ in a plane have no points in common. Then

The intercept on the line y = x by the circle x2 + y2 − 2x = 0 is AB. Equation of the circle on AB as a diameter is

Let P be situated at a distance 'd' from centre O. Which of the following does not equal the product (PQ)(PR), where T is a point on C and PT is tangent to C:

AB is a fixed chord of a circle and XY is any chord having its middle point Z on AB. Analyze: XY is greatest if XY coincides with AB.

A circle is inscribed in an equilateral triangle with side lengths 6 unit. 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:

In an acute triangle ABC, point H is the intersection point of altitude CE to AB and altitude BD to AC. A circle with DE as its diameter intersects AB and AC at points F and G respectively. If BC = 25, BD = 20 and BE = 7.Let FG and AH intersect at point K, then the length of AK =

Let L₁, L₂ and L₃ be the lengths of tangents drawn from a point P to the circles x² + y² = 4, x² + y² - 4x = 0 and x² + y² - 4y = 0 respectively. If L₁⁴ = L₂² L₃² + 16 then the locus of P are the curves, C₁ (a straight line) and C₂ (a circle).Circumcentre of the triangle formed by C₁ and two other lines which are at angle of 45° with C₁ and tangent to C₂ is:

The distance between the chords of contact of tangents to the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ from the origin and the point $(g, f)$ is:

Given a line segment AB, where A is at (0, 0) and B at (a, 0). Three circles S₁, S₂, S₃ of radius R are centred at the endpoints and the midpoint of the line segment AB. A fourth circle S₄ is drawn touching the 3 given circles.If \(0

For what values of l and m the circle $5(x^2 + y^2) + ly - m = 0$ belongs to the coaxal system determined by the circles $x^2 + y^2 + 2x + 4y - 6 = 0$ and $2(x^2 + y^2) - x = 0$?

From any point on the circle x2 + y2 = a2 tangents are drawn to the circle x2 + y2 = a2sin2 $\alpha$, the angle between them is

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 :

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

Circles Questions (554)