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Problem 19

# If $$(a, b)$$ is a point on the chord $$A B$$ of the circle, where the ends of the chord are $$A=(-2,3)$$ and $$B \equiv(3,2)$$ then : (a) $$a \in[-3,2], b \in[2,3]$$ (b) $$a \in[2,3], b \in[-3,2]$$ (c) $$a \in[-2,2], b \in[-3,3]$$ (d) $$a \in[-3,3], b \in[-2,2]$$

Expert verified
The solution is: (a) $$a \in[-2,3], b \in[2,3]$$
See the step by step solution

## Step 1 Understanding the Problem

Read the problem carefully. We are given the end points of a chord in a circle as $$A=(-2,3)$$ and $$B \equiv(3,2)$$. We need to find the range within which the points $$(a, b)$$ on the chord $$AB$$ will vary. This requires understanding of coordinate geometry and chord properties.

## Step 2 Plotting the Points

Plot the points $$A$$ and $$B$$ given in the problem. We can see that $$A$$ is situated on the left side (negative x-coordinates) and $$B$$ is on the right (positive x-coordinates). Also, the y-coordinate of $$A$$ is greater than $$B$$. Hence, a point on the chord will vary in such a way to satisfy the properties of chords in a circle and also should lie in between $$A$$ and $$B$$.

## Step 3 Finding the Range

The x-coordinate of any point on the chord will be in between -2 and 3 i.e $$a \in[-2,3]$$ as per the points $$A$$ and $$B$$. Similar logic applies for the y-coordinate. $$B$$'s y-coordinate is 2 and $$A$$'s y-coordinate is 3, so any point $$b$$ on the chord will have y-coordinate varying from 2 to 3 i.e $$b \in [2,3]$$. Hence, the point $$(a, b)$$ will be such that the x-coordinate of $$a$$ is in between -2 and 3 and the y-coordinate of $$b$$ is in between 2 and 3.

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