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4. A circle touches a g8iven straight line and cuts off a constant length 2d from another straight line perpendicular to the first straight line. The locus of the centre of the circle is

(a) Hyperbola

(b) Circle

(c) Parabola

(d) Pair of perpendicular lines

[The two straight lines to be considered as coordinate axes]

Simran Bhatia , 11 Years ago
Grade 11
anser 1 Answers
Jitender Singh

Last Activity: 10 Years ago

Ans: (a) Hyperbola
Sol:
Let the centre of circle be (h, k), it touches the x-axis and cut a constant lenght of ‘2d’
on y-axis.
Equation of circle:
(x-h)^{2}+(y-k)^{2}=k^{2}
Circle intersection with y-axis (x = 0);
(-h)^{2}+(y-k)^{2}=k^{2}
y^{2}-2hk+h^{2}=0
y = \frac{k\pm \sqrt{k^{2}-4h^{2}}}{2}
Now, equate given length with difference of intersecting coordinates
\sqrt{k^{2}-4h^{2}} = 2d
{k^{2}-4h^{2}} = 4d^{2}
h\rightarrow x, k\rightarrow y
{y^{2}-4x^{2}} = 4d^{2}
Thanks & Regards
Jitender Singh
IIT Delhi
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