Circle 1 Question 1
1. A circle touching the $X$-axis at $(3,0)$ and making a intercept of length 8 on the $Y$-axis passes through the point
(2019 Main, 12 April II)
(a) $(3,10)$
(b) $(3,5)$
(c) $(2,3)$
(d) $(1,5)$
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Answer:
Correct Answer: 1. (a)
Solution:
- It is given that the circle touches the $X$-axis at $(3,0)$ and making an intercept of 8 on the $Y$-axis.
Let the radius of the circle is ’ $r$ ‘, then the coordinates of centre of circle are $(3, r)$.
From the figure, we have
$$ \begin{array}{rlrl} C M & =3, \\ C A & =\text { radius }=r \\ \text { and } \quad & A M & =B M=\frac{A B}{2}=4 \\ \text { Then, } \quad & r^{2} & =C M^{2}+A M^{2}=9+16=25 \\ \Rightarrow \quad & r & = \pm 5 \end{array} $$
Now, the equation of circle having centre $(3, \pm 5)$ and radius $=5$ is
$$ (x-3)^{2}+(y \pm 5)^{2}=25 $$
Now, from the options $(3,10)$ satisfy the equation of circle
$$ (x-3)^{2}+(y-5)^{2}=25 $$
