Find the equation of the circle whose...

MathematicsStandard and General Equation of a CircleJEE Advanced 1978Moderate

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

Visualized Solution (Hindi)

Analyze the Given Circle

Given Circle: $x^{2} + y^{2} - 2 x - 4 y - 20 = 0$
Comparing with $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ :
Center $C_{1} (- g, - f) = (1, 2)$
Radius $r_{1} = 1^{2} + 2^{2} - (- 20) = 25 = 5$

The Geometric Condition

Required Circle: Radius $r_{2} = 5$
Point of Contact: $P (5, 5)$
Since $r_{1} = r_{2} = 5$ , $P$ is the midpoint of the line joining $C_{1} (1, 2)$ and $C_{2} (α, β)$ .

Finding the New Center .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C_{2}

Using Midpoint Formula:
$\frac{1 + α}{2} = 5 ⟹ 1 + α = 10 ⟹ α = 9$
$\frac{2 + β}{2} = 5 ⟹ 2 + β = 10 ⟹ β = 8$
New Center $C_{2} = (9, 8)$

Formulating the Final Equation

Equation of Circle: $(x - 9)^{2} + (y - 8)^{2} = 5^{2}$
Expanding: $(x^{2} - 18 x + 81) + (y^{2} - 16 y + 64) = 25$
Simplifying: $x^{2} + y^{2} - 18 x - 16 y + 145 - 25 = 0$
Final Answer: $x^{2} + y^{2} - 18 x - 16 y + 120 = 0$

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Conceptually Similar Problems

MathematicsEquation of Tangent and NormalJEE Advanced 1991Moderate

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Two circles, each of radius 5 units, touch each other at $(1, 2)$ . If the equation of their common tangent is $4 x + 3 y = 10$ , find the equation of the circles.

Answer:x^2 + y^2 + 6x + 2y - 15 = 0 \text{ and } x^2 + y^2 - 10x - 10y + 25 = 0

MathematicsLength of Tangent and Chord of ContactJEE Advanced 2012Moderate

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The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4 x - 5 y = 20$ to the circle $x^{2} + y^{2} = 9$ is

(A)

20 (x^{2} + y^{2}) - 36 x + 45 y = 0

(B)

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(C)

36 (x^{2} + y^{2}) - 20 x + 45 y = 0

(D)

36 (x^{2} + y^{2}) + 20 x - 45 y = 0

Answer:A

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Find the equation of circle touching the line $2 x + 3 y + 1 = 0$ at $(1, - 1)$ and cutting orthogonally the circle having line segment joining $(0, 3)$ and $(- 2, - 1)$ as diameter.

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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

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A circle touches the line $y = x$ at a point $P$ such that $O P = 42$ , where $O$ is the origin. The circle contains the point $(- 10, 2)$ in its interior and the length of its chord on the line $x + y = 0$ is $62$ . Determine the equation of the circle.

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MathematicsStandard and General Equation of a CircleJEE Advanced 1986Moderate

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Lines $5 x + 12 y - 10 = 0$ and $5 x - 12 y - 40 = 0$ touch a circle $C_{1}$ of diameter 6. If the centre of $C_{1}$ lies in the first quadrant, find the equation of the circle $C_{2}$ which is concentric with $C_{1}$ and cuts intercepts of length 8 on these lines.

Answer:x^2 + y^2 - 10x - 4y + 4 = 0

MathematicsStandard and General Equation of a CircleJEE Advanced 1983Easy

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The points of intersection of the line $4 x - 3 y - 10 = 0$ and the circle $x^{2} + y^{2} - 2 x + 4 y - 20 = 0$ are ......... and .........

Answer:(4, 2), (-2, -6)

MathematicsStandard and General Equation of a CircleJEE Main 2006Easy

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If the lines $3 x - 4 y - 7 = 0$ and $2 x - 3 y - 5 = 0$ are two diameters of a circle of area $49 π$ square units, the equation of the circle is

(A)

x^{2} + y^{2} + 2 x - 2 y - 47 = 0

(B)

x^{2} + y^{2} + 2 x - 2 y - 62 = 0

(C)

x^{2} + y^{2} - 2 x + 2 y - 62 = 0

(D)

x^{2} + y^{2} - 2 x + 2 y - 47 = 0

Answer:D

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

(A)

x^{2} + y^{2} + 2 x - 2 y - 23 = 0

(B)

x^{2} + y^{2} - 2 x - 2 y - 23 = 0

(C)

x^{2} + y^{2} + 2 x + 2 y - 23 = 0

(D)

x^{2} + y^{2} - 2 x + 2 y - 23 = 0

Answer:D

MathematicsDirector Circle and Family of CirclesJEE Advanced 1986Easy

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The equation of the line passing through the points of intersection of the circles $3 x^{2} + 3 y^{2} - 2 x + 12 y - 9 = 0$ and $x^{2} + y^{2} + 6 x + 2 y - 15 = 0$ is .........

Answer:10x - 3y - 18 = 0

.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).

Visualized Solution (Hindi)

Analyze the Given Circle

The Geometric Condition

Finding the New Center .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C2​

Formulating the Final Equation

Conceptually Similar Problems

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Two circles, each of radius 5 units, touch each other at (1,2). If the equation of their common tangent is 4x+3y=10, find the equation of the circles.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x−5y=20 to the circle x2+y2=9 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the equation of circle touching the line 2x+3y+1=0 at (1,−1) and cutting orthogonally the circle having line segment joining (0,3) and (−2,−1) as diameter.

.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.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The points of intersection of the line 4x−3y−10=0 and the circle x2+y2−2x+4y−20=0 are ......... and .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the lines 3x−4y−7=0 and 2x−3y−5=0 are two diameters of a circle of area 49π square units, the equation of the circle is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the lines 2x+3y+1=0 and 3x−y−4=0 lie along diameter of a circle of circumference 10π, then the equation of the circle is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation of the line passing through the points of intersection of the circles 3x2+3y2−2x+12y−9=0 and x2+y2+6x+2y−15=0 is .........

.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; } Two circles, each of radius 5 units, touch each other at (1,2). If the equation of their common tangent is 4x+3y=10, find the equation of the circles.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x−5y=20 to the circle x2+y2=9 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the equation of circle touching the line 2x+3y+1=0 at (1,−1) and cutting orthogonally the circle having line segment joining (0,3) and (−2,−1) as diameter.

.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.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The points of intersection of the line 4x−3y−10=0 and the circle x2+y2−2x+4y−20=0 are ......... and .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the lines 3x−4y−7=0 and 2x−3y−5=0 are two diameters of a circle of area 49π square units, the equation of the circle is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the lines 2x+3y+1=0 and 3x−y−4=0 lie along diameter of a circle of circumference 10π, then the equation of the circle is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation of the line passing through the points of intersection of the circles 3x2+3y2−2x+12y−9=0 and x2+y2+6x+2y−15=0 is .........