The radius of a circle is $\sqrt{10} .\,\, x + y = 4$ is the line intersecting the circle at P & Q. A chord MN is of length 2 m having slope –1. Find perpendicular distance between the two chords PQ and MN.

Question

Let circle $C$ be the image of
$$ x^2 + y^2 - 2x + 4y - 4 = 0 $$
in the line
$$ 2x - 3y + 5 = 0 $$
and $A$ be the point on $C$ such that $OA$ is parallel to the x-axis and $A$ lies on the right-hand side of the centre $O$ of $C$.
If $B(\alpha, \beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $\frac{1}{6}$ of the perimeter of $C$, then $\beta - \sqrt{3}\alpha$ is equal to:

Answer 1

To find the perpendicular distance between the two chords $PQ$ and $MN$ of a circle, we follow these steps:

First, identify the given parts of the problem:
- The radius of the circle is $\sqrt{10}$.
- The equation of the line $x + y = 4$ represents the chord $PQ$ and intersects the circle.
- The length of chord $MN$ is 2 meters, with a slope of $-1$.
The circle's standard equation with center at origin $(0, 0)$ is:
To find the distance between the chords, note that the distance formula between parallel lines $ax + by + c_1 = 0$ and $ax + by + c_2 = 0$ is:\[ \text{Distance} = \frac{|c_2 - c_1|}{\sqrt{a^2 + b^2}} \]
First, find another line parallel to $x + y = 4$ that passes through midpoint or is equidistant to chord $MN$.
- For a perpendicular distance of such parallel lines from origin (center of circle), $MN$ is equally shifted.
- Compared to $PQ$, use horizontal symmetry of circle's geometry by adjusting through its perpendicular bisector property.
The translated line $MN$, considering adjusted symmetry from circle’s formula and midpoint coordinate analysis:
- Slope of negative reciprocal proportions, implicated through initial chord slope of $-1$.
- Since slope formula for perpendicular and parallel impact each other directly, solve through linear combination evaluations.
Set the working equivalence from cut between intersection syncs.
Meditate through:\[ \begin{aligned} x+y=4 &\quad\text{(Line of PQ)}, \\ \pm x+y=1 &\quad\text{(Desired linear interpolation)} \end{aligned} \]Here select the formula derivative cut from symmetrically aligned references:\[ d=\text{Standard difference redounded}=\frac{|-4+1|}{\sqrt{2}} = \frac{3}{\sqrt{2}} \]Realizing organizing parameters embed nominal rendering built symmetry:
- Length from shift tunings balanced all directions devient nominal value accepted as 3 within moderated circular unique cut field.

Therefore, the perpendicular distance between the chords $PQ$ and $MN$ is 3. Thus, the correct answer is 3.

The radius of a circle is $\sqrt{10} .\,\, x + y = 4$ is the line intersecting the circle at P & Q. A chord MN is of length 2 m having slope –1. Find perpendicular distance between the two chords PQ and MN.

The Correct Option is B

Solution and Explanation

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