SATHEE: Circle 5 Question 12

Circle 5 Question 12

12. From the point $A (0, 3)$ on the circle $x^{2} + 4 x + (y - 3)^{2} = 0$ , a chord $A B$ is drawn and extended to a point $M$ such that $A M = 2 A B$ . The equation of the locus of $M$ is… .

(1986, 2M)

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Solution:

Given, $(x + 2)^{2} + (y - 3)^{2} = 4$

Let the coordinate be $M (h, k)$ , where $B$ is mid-point of $A$ and $M$ .

$\Rightarrow B \frac{h}{2}, \frac{k + 3}{2}$

But $A B$ is the chord of circle

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

Thus, $B$ must satisfy above equation.

$\begin{aligned} ∴ \frac{h^{2}}{4} + \frac{4 h}{2} + \frac{1}{2} (k + 3) - 3^{2} = 0 \\ \Rightarrow h^{2} + k^{2} + 8 h - 6 k + 9 = 0 \end{aligned}$

$∴$ Locus of $M$ is the circle

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

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Circle 5 Question 12

12. From the point A(0,3) on the circle x2+4x+(y−3)2=0, a chord AB is drawn and extended to a point M such that AM=2AB. The equation of the locus of M is… .

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12. From the point $A (0, 3)$ on the circle $x^{2} + 4 x + (y - 3)^{2} = 0$ , a chord $A B$ is drawn and extended to a point $M$ such that $A M = 2 A B$ . The equation of the locus of $M$ is… .