Circles

Step 1: Center of circle is (4,0).
Step 2: Required chord is perpendicular to radius.
Step 3: Equation is y=3.

Step 1: Orthogonality condition. 2(g₁g₂+f₁f₂)=c₁+c₂
Step 2: Substitute coefficients. Solve to get k=-18.

Step 1: Solve using geometry of circles.
By applying the properties of the common chord of two circles, we use the formula for the length of the chord to find that .
Thus, the correct answer is (2).

Step 1: Rewrite the equation of the circle.
The given equation of the circle is: Completing the square, we get the standard form of the circle equation.
Step 2: Find the equation of the tangents.
Using the formula for tangents from the origin to the circle, we get the equation . Final Answer:

Step 1: Angle of intersection satisfies:
Step 2: Substituting values gives .
Step 3:

Step 1: Chord length:
Step 2: Given ,
Step 3:

Step 1: Length of tangent = √(S₁-S₂).
Step 2: Evaluating for both circles gives ratio 2:3.

Step 1: Radius squared: r²=(λ²+(1-λ)²)/(4)-5
Step 2: Condition r>5⟹ r²>25.
Step 3: Solving inequality gives 16 integral values.

Step 1: Write circle in standard form: (x-1)²+(y-1)²=4 So center C(1,1), radius r=2.
Step 2: Distance of center from line x+y-3=0: d=frac|1+1-3|√(2)=(1)/(\sqrt2)
Step 3: Length of chord: 2√(r²-d²)=2√(4-\frac12)=√(14)

In an equilateral triangle inscribed in a circle of radius a, the opposite side to vertex (a,0) is a vertical line at x=-(a)/(2). Hence the equation is 2x+a=0.

Step 1: Eliminate :

Step 2: This is the standard equation of an ellipse.

Step 1: Substitute :

Step 2: Substitute :

Step 3: Solving (1) and (2) gives:

Step 4: Eccentricity:

Step 1: Centre of the circle is (2,2).
Step 2: Radius r is the distance between (2,2) and (4,5): r=√((4-2)²+(5-2)²)=√(4+9)=√(13)
Step 3: Equation of circle: (x-2)²+(y-2)²=13

Chord length =2Rsin(π/4)=√(2)R. Given chord =√(2)⟹ R=1. Area =π R²=π.

The focus of both curves is S(2,0). Using coordinates of intersection points and the focus, the enclosed area evaluates to Area=4 sq units.

Using length of tangent formula: L=√(S₁-S) Ratio simplifies to 1:2.

Radius: r²=(λ²+(1-λ)²)/(4)-5 Condition r\le5 gives 16 integer values.

Centre of circle: (1,1), radius r=√(4)=2. Distance of centre from line x+y-3=0: d=frac|1+1-3|√(2)=\frac1√(2) Chord length: 2√(r²-d²) =2√(4-\frac12) =\frac3√(3)2

N/A

Vertices of the equilateral triangle on the circle are at angles 0^∘,120^∘,240^∘. Coordinates of the other two vertices: (-(a)/(2),±frac√(3)a2) Both have the same x-coordinate, so the opposite side is: x=-(a)/(2)⟹ 2x+a=0

The radius is half the diameter: .
The area of a circle is: Thus, option (1) is correct.