SATHEE: Circle 4 Question 18

Circle 4 Question 18

18. Consider a family of circles passing through two fixed points $A (3, 7)$ and $B (6, 5)$ . Show that the chords in which the circle $x^{2} + y^{2} - 4 x - 6 y - 3 = 0$ cuts the members of the family are concurrent at a point. Find the coordinates of this point.

(1993, 5M)

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Answer:

Correct Answer: 18. $x = 2$ and $y = 23 / 3$

Solution:

The equation of the circle on the line joining the points $A (3, 7)$ and $B (6, 5)$ as diameter is

$(x - 3) (x - 6) + (y - 7) (y - 5) = 0$

and the equation of the line joining the points $A (3, 7)$ and $B (6, 5)$ is $y - 7 = \frac{7 - 5}{3 - 6} (x - 3)$

$\Rightarrow 2 x + 3 y - 27 = 0$

Now, the equation of family of circles passing through the point of intersection of Eqs. (i) and (ii) is

$S + λ P = 0$

$\Rightarrow (x - 3) (x - 6) + (y - 7) (y - 5) + λ (2 x + 3 y - 27) = 0$

$\Rightarrow x^{2} - 6 x - 3 x + 18 + y^{2} - 5 y - 7 y + 35$ $+ 2 λ x + 3 λ y - 27 λ = 0$

$\Rightarrow S_{1} \equiv x^{2} + y^{2} + x (2 λ - 9) + y (3 λ - 12)$

$+ (53 - 27 λ) = 0$

Again, the circle,which cuts the members of family of circles, is

$S_{2} \equiv x^{2} + y^{2} - 4 x - 6 y - 3 = 0$

and the equation of common chord to circles $S_{1}$ and $S_{2}$ is

$\begin{aligned} S_{1} - S_{2} & = 0 \\ \Rightarrow & x (2 λ - 9 + 4) + y (3 λ - 12 + 6) + (53 - 27 λ + 3) & = 0 \\ \Rightarrow & 2 λ x - 5 x + 3 λ y - 6 y + 56 - 27 λ & = 0 \\ \Rightarrow & (- 5 x - 6 y + 56) + λ (2 x + 3 y - 27) & = 0 \end{aligned}$

which represents equations of two straight lines passing through the fixed point whose coordinates are obtained by solving the two equations

$5 x + 6 y - 56 = 0$ and $2 x + 3 y - 27 = 0$ ,

we get $x = 2$ and $y = 23 / 3$

Circle 4 Question 18

18. Consider a family of circles passing through two fixed points $A (3, 7)$ and $B (6, 5)$ . Show that the chords in which the circle $x^{2} + y^{2} - 4 x - 6 y - 3 = 0$ cuts the members of the family are concurrent at a point. Find the coordinates of this point.

Answer:

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Circle 4 Question 18

18. Consider a family of circles passing through two fixed points A(3,7) and B(6,5). Show that the chords in which the circle x2+y2−4x−6y−3=0 cuts the members of the family are concurrent at a point. Find the coordinates of this point.

Answer:

Engineering Preparation

Medical Preparation

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Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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18. Consider a family of circles passing through two fixed points $A (3, 7)$ and $B (6, 5)$ . Show that the chords in which the circle $x^{2} + y^{2} - 4 x - 6 y - 3 = 0$ cuts the members of the family are concurrent at a point. Find the coordinates of this point.