Equation of a circle through the origin and belonging to the co-axial system, of which the limiting points are $(1, 2), (4, 3)$ is

x^{2} + y^{2} - 2 x + 4 y = 0

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x^{2} + y^{2} - 8 x - 6 y = 0

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2 x^{2} + 2 y^{2} - x - 7 y = 0

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x^{2} + y^{2} - 6 x - 10 y = 0

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Solution

The correct option is D $2 x^{2} + 2 y^{2} - x - 7 y = 0$
Since the limiting points of a system of coaxial circles are the point circles (radius being zero), two members of the system are
$(x - 1)^{2} + (y - 2)^{2} = 0 \Rightarrow x^{2} + y^{2} - 2 x - 4 y + 5 = 0$
and
$(x - 4)^{2} + (y - 3)^{2} = 0 \Rightarrow x^{2} + y^{2} - 8 x - 6 y + 25 = 0$
The co-axial system of circles with these as members is
$x^{2} + y^{2} - 2 x - 4 y + 5 + λ (x^{2} + y^{2} - 8 x - 6 y + 25) = 0$
It passes through the origin if $5 + 25 λ = 0$ or $λ = - (1 / 5),$ which gives the equation of the required circle as
$5 (x^{2} + y^{2} - 2 x - 4 y + 5) - (x^{2} + y^{2} - 8 x 6 y + 25) = 0$
$\Rightarrow 4 x^{2} + 4 y^{2} - 2 x - 14 y = 0 \Rightarrow 2 x^{2} + 2 y^{2} - x - 7 y = 0$

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Similar questions

Q. Assertion :Statement-1: Equation of a circle through the origin and belonging to the co-axial system.of which the limiting points are

(1, 1)

and

(3, 3)

2 x^{2} + 2 y^{2} - 3 x - 3 y = 0

Reason: Equation of a circle passing through the points

(1, 1)

and

(3, 3)

x^{2} + y^{2} - 2 x - 6 y + 6 = 0

Q. The equation of the circle concentric with the circle

x^{2} + y^{2} + 8 x + 10 y - 7 = 0

and passing through the centre of the circle

x^{2} + y^{2} - 4 x - 6 y = 0

Tangents are drawn from the point P(1, 8) to the circle $x^{2} + y^{2} - 6 x - 4 y - 11 = 0$ touch the circle at the point A and B, then equation of the circumcircle of the triangle PAB is

Q. One of the limiting points of the coaxial system of circles containing the circle

x^{2} + y^{2} - 6 x - 6 y + 4 = 0

and

x^{2} + y^{2} - 2 x - 4 y + 3 = 0

is:

Q. Tangents drawn from the point P(1, 8) to the circle

x^{2} + y^{2} - 6 x - 4 y - 11 = 0

touch the circle at points A and B. The equation of the circumcircle of triangle PAB is

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Equation of a circle through the origin and belonging to the co-axial system, of which the limiting points are (1,2),(4,3) is

Equation of a circle through the origin and belonging to the co-axial system, of which the limiting points are $(1, 2), (4, 3)$ is