Circles Questions

MathematicsCircles106 Questions Found

MathematicsEquation of Tangent and NormalJEE Advanced 2008Difficult

Comprehension Passage

A circle

C

of radius 1 is inscribed in an equilateral triangle

P QR

. The points of contact of

C

with the sides

P Q, QR, R P

are

D, E, F

, respectively. The line

P Q

is given by the equation

3 x + y - 6 = 0

and the point

D

(\frac{3 3}{2}, \frac{3}{2})

. Further, it is given that the origin and the centre of

C

are on the same side of the line

P Q

Question 1:

The equation of circle $C$ is

(A)

(x - 23)^{2} + (y - 1)^{2} = 1

(B)

(x - 23)^{2} + (y + \frac{1}{2})^{2} = 1

(C)

(x - 3)^{2} + (y + 1)^{2} = 1

(D)

(x - 3)^{2} + (y - 1)^{2} = 1

Answer:D

Question 2:

Points $E$ and $F$ are given by

(A)

(\frac{3}{2}, \frac{3}{2}), (3, 0)

(B)

(\frac{3}{2}, \frac{1}{2}), (3, 0)

(C)

(\frac{3}{2}, \frac{3}{2}), (\frac{3}{2}, \frac{1}{2})

(D)

(\frac{3 3}{2}, \frac{3}{2}), (\frac{3}{2}, \frac{1}{2})

Answer:A

Question 3:

Equations of the sides $QR, R P$ are

(A)

y = \frac{2}{3} x + 1, y = - \frac{2}{3} x - 1

(B)

y = \frac{1}{3} x, y = 0

(C)

y = \frac{3}{2} x + 1, y = - \frac{3}{2} x - 1

(D)

y = 3 x, y = 0

Answer:D

MathematicsEquation of Tangent and NormalJEE Advanced 2012Moderate

View in:English Hindi

Comprehension Passage

A tangent

P T

is drawn to the circle

x^{2} + y^{2} = 4

at the point

P (3, 1)

. A straight line

L

, perpendicular to

P T

is a tangent to the circle

(x - 3)^{2} + y^{2} = 1

Question 1:

A possible equation of $L$ is

(A)

x - 3 y = 1

(B)

x + 3 y = 1

(C)

x - 3 y = - 1

(D)

x + 3 y = 5

Answer:A

Question 2:

A common tangent of the two circles is

(A)

x = 4

(B)

y = 2

(C)

x + 3 y = 4

(D)

x + 22 y = 6

Answer:D

MathematicsStandard and General Equation of a CircleJEE Advanced 1978Moderate

View in:English Hindi

Sketch the solution set of the following system of inequalities: $x^{2} + y^{2} - 2 x \leq 0; 3 x - y - 12 \leq 0; y - x \leq 0; y \geq 0$ .

MathematicsStandard and General Equation of a CircleJEE Advanced 1983Moderate

View in:English Hindi

Through a fixed point $(h, k)$ secants are drawn to the circle $x^{2} + y^{2} = r^{2}$ . Show that the locus of the mid-points of the secants intercepted by the circle is $x^{2} + y^{2} = h x + k y$ .

MathematicsStandard and General Equation of a CircleJEE Advanced 1987Moderate

View in:English Hindi

Let a given line $L_{1}$ intersects the $x$ and $y$ axes at $P$ and $Q$ , respectively. Let another line $L_{2}$ , perpendicular to $L_{1}$ , cut the $x$ and $y$ axes at $R$ and $S$ , respectively. Show that the locus of the point of intersection of the lines $P S$ and $QR$ is a circle passing through the origin.

MathematicsStandard and General Equation of a CircleJEE Advanced 1989Moderate

View in:English Hindi

If $m_{i}, \frac{1}{m _{i}}, m_{i} > 0, i = 1, 2, 3, 4$ are four distinct points on a circle, then show that $m_{1} m_{2} m_{3} m_{4} = 1$ .

MathematicsStandard and General Equation of a CircleJEE Advanced 1997Moderate

View in:English Hindi

Let $C$ be any circle with centre $(0, 2)$ . Prove that at the most two rational points can be there on $C$ . (A rational point is a point both of whose coordinates are rational numbers.)

MathematicsEquation of Tangent and NormalJEE Advanced 1998Moderate

View in:English Hindi

$C_{1}$ and $C_{2}$ are two concentric circles, the radius of $C_{2}$ being twice that of $C_{1}$ . From a point $P$ on $C_{2}$ , tangents $P A$ and $P B$ are drawn to $C_{1}$ . Prove that the centroid of the triangle $P A B$ lies on $C_{1}$ .

MathematicsStandard and General Equation of a CircleJEE Advanced 1978Moderate

View in:English Hindi

Find the equation of the circle whose radius is 5 and which touches the circle $x^{2} + y^{2} - 2 x - 4 y - 20 = 0$ at the point $(5, 5)$ .

Answer:x^2 + y^2 - 18x - 16y + 120 = 0

MathematicsStandard and General Equation of a CircleJEE Advanced 1982Moderate

View in:English Hindi

Find the equations of the circle passing through $(- 4, 3)$ and touching the lines $x + y = 2$ and $x - y = 2$ .

Answer:x^2 + y^2 + 2(10 \pm \sqrt{54})x + 55 \pm \sqrt{54} = 0

...11

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation of circle C is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Points E and F are given by

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Equations of the sides QR,RP are

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A possible equation of L is

A common tangent of the two circles is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Sketch the solution set of the following system of inequalities: x2+y2−2x≤0;3x−y−12≤0;y−x≤0;y≥0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Through a fixed point (h,k) secants are drawn to the circle x2+y2=r2. Show that the locus of the mid-points of the secants intercepted by the circle is x2+y2=hx+ky.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If mi​,mi​1​,mi​>0,i=1,2,3,4 are four distinct points on a circle, then show that m1​m2​m3​m4​=1.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the equation of the circle whose radius is 5 and which touches the circle x2+y2−2x−4y−20=0 at the point (5,5).

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the equations of the circle passing through (−4,3) and touching the lines x+y=2 and x−y=2.

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation of circle C is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Points E and F are given by

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Equations of the sides QR,RP are

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A possible equation of L is

A common tangent of the two circles is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Sketch the solution set of the following system of inequalities: x2+y2−2x≤0;3x−y−12≤0;y−x≤0;y≥0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Through a fixed point (h,k) secants are drawn to the circle x2+y2=r2. Show that the locus of the mid-points of the secants intercepted by the circle is x2+y2=hx+ky.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If mi​,mi​1​,mi​>0,i=1,2,3,4 are four distinct points on a circle, then show that m1​m2​m3​m4​=1.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the equation of the circle whose radius is 5 and which touches the circle x2+y2−2x−4y−20=0 at the point (5,5).

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the equations of the circle passing through (−4,3) and touching the lines x+y=2 and x−y=2.

The equation of circle $C$ is

Points $E$ and $F$ are given by

Equations of the sides $QR, R P$ are

A possible equation of $L$ is

Sketch the solution set of the following system of inequalities: $x^{2} + y^{2} - 2 x \leq 0; 3 x - y - 12 \leq 0; y - x \leq 0; y \geq 0$ .

Through a fixed point $(h, k)$ secants are drawn to the circle $x^{2} + y^{2} = r^{2}$ . Show that the locus of the mid-points of the secants intercepted by the circle is $x^{2} + y^{2} = h x + k y$ .

If $m_{i}, \frac{1}{m _{i}}, m_{i} > 0, i = 1, 2, 3, 4$ are four distinct points on a circle, then show that $m_{1} m_{2} m_{3} m_{4} = 1$ .

Find the equation of the circle whose radius is 5 and which touches the circle $x^{2} + y^{2} - 2 x - 4 y - 20 = 0$ at the point $(5, 5)$ .

Find the equations of the circle passing through $(- 4, 3)$ and touching the lines $x + y = 2$ and $x - y = 2$ .

The equation of circle $C$ is

Points $E$ and $F$ are given by

Equations of the sides $QR, R P$ are

A possible equation of $L$ is

Sketch the solution set of the following system of inequalities: $x^{2} + y^{2} - 2 x \leq 0; 3 x - y - 12 \leq 0; y - x \leq 0; y \geq 0$ .

Through a fixed point $(h, k)$ secants are drawn to the circle $x^{2} + y^{2} = r^{2}$ . Show that the locus of the mid-points of the secants intercepted by the circle is $x^{2} + y^{2} = h x + k y$ .

If $m_{i}, \frac{1}{m _{i}}, m_{i} > 0, i = 1, 2, 3, 4$ are four distinct points on a circle, then show that $m_{1} m_{2} m_{3} m_{4} = 1$ .

Find the equation of the circle whose radius is 5 and which touches the circle $x^{2} + y^{2} - 2 x - 4 y - 20 = 0$ at the point $(5, 5)$ .

Find the equations of the circle passing through $(- 4, 3)$ and touching the lines $x + y = 2$ and $x - y = 2$ .