First slide

Definition of a circle

Question

If $(2, 4)$ is a point interior to the circle $x^{2} + y^{2} - 6 x - 10 y + λ = 0$ and the circle does not cut the axes at any point, then

Moderate

A

$λ \in (25, 32)$
B

$λ \in (9, 32)$
C

$λ \in (32, \infty)$
D

none of these

Solution

We have,
$x^{2} + y^{2} - 6 x - 10 y + λ = 0$ …(i)
The circle does not cut the coordinate axes. Therefore,
$9 - λ < 0$ and $25 - λ < 0$ [Using $: g^{2} - c < 0, f^{2} - c < 0$ ]
$\Rightarrow λ > 25 . . . (i i)$
It is given that the point $(2, 4)$ lies inside the circle (i).
$∴ 4 + 16 - 12 - 40 + λ < 0 \Rightarrow λ < 32 \dots (iii)$
From (ii) and (iii), we get $λ \in (25, 32) . . . (i i i)$

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