9th Class Math Chapter 11 Notes

Q1 — Kashif's possible locations: $6.8$ km from $A$, $7.3$ km from $B$ ($AB=8$ km)

Mark the points where Kashif could live.

1. Locus 1: circle of radius $6.8$ km centred at $A$. Locus 2: circle of radius $7.3$ km centred at $B$.
2. Since $6.8+7.3=14.1>8$ and $|7.3-6.8|=0.5<8$, the two circles intersect at exactly two points.
3. Placing $A=(0,0)$, $B=(8,0)$ and solving $x^2+y^2=46.24$, $(x-8{)}^2+y^2=53.29$ gives $x\approx 3.56$, $y\approx \pm 5.79$.

Answer: Two possible points, at $\approx (3.56,5.79)$ and $\approx (3.56,-5.79)$ relative to $A$ — the two intersections of the two circles

Q2 — Locus equidistant from two cell towers $A,B$

Draw the locus of points equidistant from $A$ and $B$.

1. The set of points equidistant from two fixed points is, by definition, the perpendicular bisector of the segment joining them.

Answer: A straight line: the perpendicular bisector of $\overline{AB}$

Q3 — Quarantine zone: locus $10$ km from a source point

Draw the locus of points $10$ km from the infection source.

1. The set of points a fixed distance from a single point is a circle of that radius centred at the point.

Answer: A circle of radius $10$ km centred at the source

Q4 — Treasure: $24$ km from $A$, equidistant from $B$ and $C$

Using scale $1$ cm $=10$ km, find where the treasure could be buried.

1. Locus 1: circle of radius $24$ km ($2.4$ cm at scale) centred at $A$.
2. Locus 2: the perpendicular bisector of $\overline{BC}$.
3. The treasure lies at the point(s) where this circle and this perpendicular bisector intersect.

Answer: Intersection point(s) of the $24$ km circle around $A$ with the perpendicular bisector of $BC$

Q5 — Mango tree: $64$ m from apple tree, $54$–$82$ m from banana tree ($90$ m apart)

Using scale $1$ cm $=10$ m, find the points where the mango tree should be planted.

1. Locus 1: circle of radius $64$ m ($6.4$ cm) centred at the apple tree.
2. Locus 2: the annular region between two circles centred at the banana tree — inner radius $54$ m ($5.4$ cm), outer radius $82$ m ($8.2$ cm).
3. $M$ must lie on the arc of the $64$ m circle that falls inside this annulus.

Answer: The arc of the $6.4$ cm circle (apple tree) that lies between the $5.4$ cm and $8.2$ cm circles (banana tree)

Q6 — Locus $32$ m from house $Q$ ($QR=73$ m)

Using scale $1$ cm $=10$ m, construct the locus of $P$ at distance $32$ m from $Q$.

1. A fixed distance from a point traces a circle.

Answer: A circle of radius $32$ m ($3.2$ cm) centred at $Q$

Q7 — Locus $48$ m from line $QR$

Construct the locus of $P$ at distance $48$ m from the line joining $Q$ and $R$.

1. A fixed distance from an infinite line traces two parallel lines, one on each side.

Answer: Two lines parallel to $QR$, each $48$ m ($4.8$ cm) away, one on either side

Q8 — Field $ABCD$, $AB=70$ m, $BC=60$ m: region near $C$, far from $AB$

Using scale $1$ cm $=10$ m, shade the region inside the field that is less than $30$ m from $C$ and farther than $25$ m from $\overline{AB}$.

1. Place $A=(0,0)$, $B=(70,0)$, $C=(70,60)$, $D=(0,60)$ (metres).
2. "Less than $30$ m from $C$": inside the circle $(x-70{)}^2+(y-60{)}^2<900$ ($3$ cm radius at scale).
3. "Farther than $25$ m from $AB$": the region $y>25$ (beyond the line $2.5$ cm above $AB$ at scale).

Answer: The region inside the rectangle satisfying both $y>25$ m and within $30$ m of corner $C$ — a rounded region near $C$, above the $25$ m line