Parabola 1 Question 2

2. A circle cuts a chord of length $4 a$ on the $X$-axis and passes through a point on the $Y$-axis, distant $2 b$ from the origin. Then, the locus of the centre of this circle, is

(2019 Main, 11 Jan, II)

(a) a parabola

(b) an ellipse

(c) a straight line

(d) a hyperbola

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Answer:

Correct Answer: 2. (a)

Solution:

  1. According to given information, we have the following figure.

Let the equation of circle be

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

According the problem,

$$ 4 a=2 \sqrt{g^{2}-c} $$

$[\because$ The length of intercepts made by the circle $x^{2}+y^{2}+2 g x+2 f y+c=0$

$$ \text { with } \left.X \text {-axis is } 2 \sqrt{g^{2}-c}\right] $$

Also, as the circle is passing through $P(0,2 b)$

$$ \begin{array}{lrr} \therefore & 0+4 b^{2}+0+4 b f+c=0 & \text { [using Eq. (i)] } \\ \Rightarrow & 4 b^{2}+4 b f+c=0 & \ldots \text { (iii) } \end{array} $$

Eliminating ’ $c$ ’ from Eqs. (ii) and (iii), we get

$$ \begin{aligned} & 4 b^{2}+4 b f+g^{2}-4 a^{2}=0 \\ & {\left[\because 4 a=2 \sqrt{g^{2}-c} \Rightarrow c=g^{2}-4 a^{2}\right] } \end{aligned} $$

So, locus of $(-g,-f)$ is

$$ \begin{array}{rlrl} & & 4 b^{2}-4 b y+x^{2}-4 a^{2}=0 \\ \Rightarrow & x^{2}=4 b y+4 a^{2}-4 b^{2} \end{array} $$

which is a parabola.



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