Mathematics - Circle Question with Solution | TestHub

MathematicsCircleGeneral, Basic geometries, Definition Diametrical form of circleEasy2 minPYQ_2013

MathematicsEasymultiple choice

Circle(s) touching $x$ -axis at a distance $3$ from the origin and having an intercept of length $2 \sqrt{7}$ on $y$ -axis is (are)

Options:(select one or more)

Answer:

A, C

Solution:

Let the circle be

$x^{2} + y^{2} + 2 g x + 2 f y + c = 0$

According to the question,

$g^{2} - c = 0 \Rightarrow g^{2} = c . . . . (i)$

and

$2 \sqrt{f^{2} - c} = 2 \sqrt{7} \Rightarrow f^{2} - c = 7 . . . . . (i i)$

The circle passes through the point $(3, 0)$ , then
$9 + 0 + 6 g + 0 + c = 0$
$\Rightarrow 9 + 6 g + g^{2} = 0,$ by $(i)$
$\Rightarrow (g + 3)^{2} = 0 \Rightarrow g = - 3$
$∴$ $c = 9$ and $f^{2} = 16 \Rightarrow f = \pm 4$

So the required circles are

$x^{2} + y^{2} - 6 x \pm 8 y + 9 = 0$ .

Stream:JEE_ADVSubject:MathematicsTopic:CircleSubtopic:General, Basic geometries, Definition Diametrical form of circle

⏱ 2mℹ️ Source: PYQ_2013

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