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  • A variable circle passes through the point A (a, b) and touches the x - axis. The locus of the other end of the diameter through A is  <img alt="(\mathrm{x}-\mathrm{a})^2=\mathrm{kby}" src="ht

A variable circle passes through the point A (a, b) and touches the x - axis. The locus of the other end of the diameter through A is  (\mathrm{x}-\mathrm{a})^2=\mathrm{kby}, where k=

Option: 1

1


Option: 2

2


Option: 3

3


Option: 4

4


Answers (1)

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Let the other end of the diameter be \mathrm{e}\left(\mathrm{x}_{\mathrm{l}}, \mathrm{y}_1\right) , The centre and the radius of the required circle are given by 

\mathrm{\begin{aligned} & C\left(\frac{x_l+a}{2}, \frac{y_l+b}{2}\right) \text { and Radius }=\frac{y_l+b}{2} \\ & \Rightarrow\left(\frac{x_l+a}{2}-a\right)^2+\left(\frac{y_l+b}{2}-b\right)^2=\left(\frac{y_l+b}{2}\right)^2 \\ & \Rightarrow\left(\frac{x_l-a}{2}\right)^2+\left(\frac{y_l-b}{2}\right)^2=\left(\frac{y_l+b}{2}\right)^2 \\ & \Rightarrow x_l^2+a^2-2 a x_l-2 b y_l=2 b y_l \\ & \Rightarrow x_l^2+a^2-2 a x_l-4 b y_l=0 \end{aligned)}

\mathrm{\Rightarrow x^2+a^2-2 a x-4 b y=0} Hence the locus of  \mathrm{\left(x_1, y_1\right)} is \mathrm{\Rightarrow(x-a)^2=4 b y}

 

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Ritika Jonwal

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