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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:

A) \( \Large x^{2}+y^{2}-xx_{1}-yy_{1}=0 \)
B) \( \Large x^{2}+y^{2}=x^{2}_{1}+y^{2}_{1} \)
C) \( \Large x+y=x_{1}+y'_{2} \)
D) \( \Large x+y=x^{2}_{1}+y^{2}_{1} \)

Correct Answer:

A) \( \Large x^{2}+y^{2}-xx_{1}-yy_{1}=0 \)

Description for Correct answer:

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

\( \Large x^{2}+y^{2}=a^{2}\ is\ xx_{1}+yy_{1}=a^{2} \)

\( \Large \therefore\ x^{2}+y^{2} = a^{2} \left(\frac{xx_{1}+yy_{1}}{a^{2}}\right) \)

=> \( \Large x^{2}+y^{2}-xx_{1}-yy_{1}=0 \)

Part of solved Circles questions and answers : >> Elementary Mathematics >> Circles

Comments

Similar Questions

1). The equations of the tangents to the circle \( \Large x^{2}+y^{2}-6x+4y-12=0 \) which are parallel to the line \( \Large 4x+3y+5=0 \),are

A). \( \Large 4x+3y+11=0 \) and \( \Large 4x+3y+8=0 \)

B). \( \Large 4x+3y-9=0 \) and \( \Large 4x+3y+7=0 \)

C). \( \Large 4x+3y+19=0 \) and \( \Large 4x+3y-31=0 \)

D). \( \Large 4x+3y-10=0 \) and \( \Large 4x+3y+12=0 \)

2). The lines \( \Large 2x-3y=5\ and\ 3x-4y=7 \) are diameters of a circle having area as 154 sq unit. Then the equation of the circle is:

A). \( \Large x^{2}+y^{2}+2x-2y=62 \)

B). \( \Large x^{2}+y^{2}+2x-2y=47 \)

C). \( \Large x^{2}+y^{2}-2x+2y=47 \)

D). \( \Large x^{2}+y^{2}-2x+2y=62 \)

3). The pole of the straight line \( \Large x+2y=1 \) with respect to the circle \( \Large x^{2}+y^{2}=5 \) is:

A). \( \Large \left(5,\ 5\right) \)

B). \( \Large \left(5,\ 10\right) \)

C). \( \Large \left(10,\ 5\right) \)

D). \( \Large \left(10,\ 10\right) \)

4). If the circle \( \Large x^{2}+y^{2}+6x-2y+k=0 \) bisects the circumference of the circle \( \Large x^{2}+y^{2}+2x-6y-15=0 \) then k is equal to:

A). 21

B). -21

C). 23

D). -23

5). If \( \Large 5x-12y=10 \) and \( \Large 12y-5x+16=0 \) are two tangents to a circle, then the radius of the circle is

A). 1

B). 2

C). 4

D). 6

6). Equation of the circle passing through the point \( \Large \left(3,\ 4\right) \) and concentric with the circle \( \Large x^{2}+y^{2}-2x-4y+1=0 \) is

A). \( \Large x^{2}+y^{2}-2x-4y=0 \)

B). \( \Large x^{2}+y^{2}-2x-4y+3=0 \)

C). \( \Large x^{2}+y^{2}-2x-4y-3=0 \)

D). none of the above

7). Circle \( \Large x^{2}+y^{2}-2x-nx-1=0 \) passes through two fixed points, co-ordinates of the points are

A). \( \Large \left(0,\ \pm 1\right) \)

B). \( \Large \left(\pm\ 1,\ 0\right) \)

C). \( \Large \left(0,\ 1\right)\ and\ \left(0,\ 2\right) \)

D). \( \Large \left(0,\ -1\right)\ and\ \left(0,\ -2\right) \)

8). Centre of circle whose normals are \( \Large x^{2}-2xy-3x+6y=0 \) is

A). \( \Large \left(3,\ \frac{3}{2}\right) \)

B). \( \Large \left(3,\ -\frac{3}{2}\right) \)

C). \( \Large \left(\frac{3}{2},\ 3\right) \)

D). none of these

9). The locus of centre of a circle \( \Large x^{2}+y^{2}-2x-2y+1=0 \) which rolls outside the circle \( \Large x^{2}+y^{2}-6x+8y=0 \) is:

A). \( \Large x^{2}+y^{2}-2x-2y-34=0 \)

B). \( \Large x^{2}+y^{2}-6x+8y+11=0 \)

C). \( \Large x^{2}+y^{2}-6x+8y-11=0 \)

D). none of these

10). A line through \( \Large P \left(1,\ 4\right) \) intersect a circle \( \Large x^{2}+y^{2}=16 \) at A and B, then PA-PB is equal to:

A). 1

B). 2

C). 3

D). 4