>> Elementary Mathematics >> Circles

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Contents:

- Elementary Mathematics
- Area and perimeter
- Circles
- Clocks
- Factorisation
- Geometry
- Height and Distance
- Indices and Surd
- LCM and HCF
- Loci and concurrency
- Logarithms
- Polynomials
- Quadratic Equations
- Quadrilateral and parallelogram
- Rational expression
- Real Analysis
- Rectangular and Cartesian products
- Set theory
- Simple and Decimal fraction
- Simplification
- Statistics
- Straight lines
- Triangle
- Trigonometric ratio
- Trigonometry
- Volume and surface area

1). The square of the length of the tangent from \( \Large \left(3,\ -4\right) \) to the circle \( \Large x^{2}+y^{2}-4x-6y+3=0 \) is:
Length of tangent from the point \( \Large \left(x_{1},y_{1}\right) \) to the circle \( \Large x^{2} + y^{2} +2gx + 2fy + c = 0 \) is \( \Large \sqrt{x_{1}^{2} +y_{1}^{2} + 2gx_{1} + 2fy_{1} + c } \) \( \Large \sqrt{3^{2} + 4^{2} + -4(3) - 6(-4) + 3} = \sqrt{40}\) | ||||

2). If \( \Large g^{2}+f^{2}=c \) then the equation \( \Large x^{2}+y^{2}+2gx+2fy+c=0 \) will represent:
Given that \( \Large x^{2}+y^{2}+2gx+2fy+c=0 \) and \( \Large g^{2}+f^{2}=c \) Radius of circle = \( \Large \sqrt{g^{2}+f^{2}-c} \) \( \Large \therefore g^{2}+f^{2}=c \) => Radius = 0 Thus given equation represents a circle of radius 0. | ||||

3). The limit of the perimeter of the regular n polygons inscribe in a circle of radius R as \( \Large n\ \rightarrow\ \infty \) is:
As \( \Large n\ \rightarrow\ \infty \), therefore polygon becomes a circle and perimeter of circle = \( \Large 2 \pi R \). | ||||

4). The value of n, for which the circle \( \Large x^{2}+y^{2}+2nx+6y+1=0 \) intersects the circle \( \Large x^{2}+y^{2}+4x+2y=0 \) orthogonally is:
Equations of circles are | ||||

5). The Value of c for which the line \( \Large y=2x+c \) is a tangent to the circle \( \Large x^{2}+y^{2}=16 \) is:
Given that \( \Large y = 2x+c \) ...(i) | ||||

6). The radical axis of two circle and line joining their centres are:
Radical axis is the common chord of the two circles and radical axis is perpendicular to the line joining the centers of two circles. | ||||

7). Which of the following is a point on the common chord of the circles \( \Large x^{2}+y^{2}+2x-3y+6=0\ and\ x^{2}+y^{2}+x-8y-13=0 \)?
Let the equation of circles are \( \Large S_{1} = x^{2}+y^{2}+2x-3y+6=0 \) ...(i) \( \Large S_{2} = x^{2}+y^{2}+x-8y-13=0 \) ...(ii) Equation of common chord is \( \Large S_{1} - S_{2} = 0 \) => \( \Large \left(x^{2}+y^{2}+2x-3y+6\right) - \left(x^{2}+y^{2}+x-8y-13\right) = 0 \) => \( \Large x+5y+19=0 \) ...(iii) In the given options only the point \( \Large \left(1,\ -4\right) \) satisfied the eq. (iii) | ||||

8). The radius of the circle passing through the point \( \Large \left(6,\ 2\right) \) and two of whose diameter are \( \Large x+y=6\ and\ x+2y=4 \) is:
Centre is the point of intersection of two diameters i.e., the point of intersection of two diameter is \( \Large C \left(8,\ -2\right) \) therefore the distance from the centre to the point \( \Large P \left(6,\ 2\right) \) is \( \Large r = CP = \sqrt{4+16} = \sqrt{20} \) | ||||

9). The radius of any circle touching the lines \( \Large 3x-4y+5=0\ and\ 6x-8y-9=0 \) is:
Since, given lines are parallel to each other so the line segment joining the points of contact is diameter of the circle, distance between the lines | ||||

10). The locus of the middle point of the chords of the circle \( \Large x^{2}+y^{2}=a^{2} \) such that the chords pass through a given point \( \Large \left(x_{1},\ y_{1}\right) \) is:
Let \( \Large P \left(x_{1}, y_{1}\right) \) be the point, then the chord of contact of tangents drawn from P to the circle |