First slide

Definition of a circle

Question

A straight line moves such that the algebraic sum of the perpendicular drawn to it from two points is equal to 2k. then the straight line always touches a fixed circle of radius

Moderate

A

2k
B

k /2
C

k
D

3k

Solution

Let the coordinates be $A (a, 0)$ and $B (- a, 0)$ and let the straight line be $y = m x + c$ then
$\frac{m x + c}{\sqrt{1 + m^{2}}} + \frac{- m x + c}{\sqrt{1 + m^{2}}} = 2 k$

$\Rightarrow c = k \sqrt{1 + m^{2}}$

So the straight lines $y = m x + k \sqrt{1 + m^{2}}$

Clearly, it touches the circle $x^{2} + y^{2} = k^{2}$ of radius k.

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