FBISE, NCP & NBF-aligned resources for Classes 9–12
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Math Chapter 9 : Topic 4
Loci
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A mobile tower must be installed $5\ km$ from a village represented by point $A$. The possible locations form:
Reasoning: Fixed distance from a point $\Longrightarrow$ circle.
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A pipeline must be laid $3\ m$ away from a straight road. The locus of all such points is:
Reasoning: Points at fixed distance from a line lie on two parallel lines.
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Two cities $A$ and $B$ want a hospital equidistant from both. The correct geometric location is:
Reasoning: Equidistant from two points $\Longrightarrow$ perpendicular bisector.
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A point is equidistant from two intersecting roads. The locus is:
Reasoning: Equidistant from two lines $\Longrightarrow$ angle bisectors.
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A point is $4\ cm$ from $A$ and $4\ cm$ from $B$. The possible number of such points is:
Reasoning: Two circles intersect at two points.
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A point is $6\ cm$ from a point and $6\ cm$ from a line. The number of possible solutions is generally:
Reasoning: Circle intersects parallel lines at two points.
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If two loci intersect at no point, it indicates:
Reasoning: No intersection $\Longrightarrow$ no common point satisfies both conditions.
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A point is equidistant from two parallel lines. The locus is:
Reasoning: Equal distance from parallel lines $\Longrightarrow$ middle line.
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A communication tower must be $8\ km$ from city $A$ and equidistant from cities $B$ and $C$. Required construction:
Reasoning: Distance from $A\ \ \ \ \Longrightarrow \ \ \ \ $circle ; equidistant from $B\ \&\ C\ \ \ \Longrightarrow$ perpendicular bisector.
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A point lies at equal distance from two intersecting lines. Which statement is correct?
Reasoning: Definition of angle bisector.
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A water tank must be located $10\ m$ from road $l$ and $10\ m$ from road $m$ where $l \parallel m$. The locus is:
Reasoning: Equal distance from parallel lines $\Longrightarrow$ central line.
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A point is $5\ cm$ from line $l$. How many such loci exist?
Reasoning: Two parallel lines at distance $5\ cm$.
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Which locus represents all points equidistant from $A$ and $B$?
Reasoning: Standard locus definition.
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Two loci intersect at exactly one point when:
Reasoning: Tangency $\Longrightarrow$ single solution.
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A point is $7\ cm$ from $A$ and lies on the perpendicular bisector of $AB$. Possible number of positions:
Reasoning: Circle intersects bisector at two points.
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A point is equidistant from three non-collinear points $A$, $B$ and $C$. It lies at:
Reasoning: Circumcenter is equidistant from vertices.
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A location must be $5\ km$ from road $l$ and $5\ km$ from road $m$ where they intersect. The locus is:
Reasoning: Equal distance from intersecting lines $\Longrightarrow$ angle bisectors.
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A point satisfies only one condition of two loci. This means:
Reasoning: Must satisfy both conditions.
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A farmer wants a well equidistant from two fences that intersect. The correct position lies on:
Reasoning: Equidistant from lines $\Longrightarrow$ angle bisector.
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A point is $3\ cm$ from $A$ and $4\ cm$ from $B$. The construction involves:
Reasoning: Two circles centered at $A$ and $B$.
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Which pair of loci can never intersect?
Reasoning: Parallel lines never meet.
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A point lies $6\ cm$ from $A$ and equidistant from two intersecting lines. Number of possible solutions is usually:
Reasoning: Circle intersects two angle bisectors $\Longrightarrow$ up to $4$ points.
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If two loci coincide completely, number of solutions is:
Reasoning: Same locus $\Longrightarrow$ infinite common points.
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A point lies on angle bisector and is $5\ cm$ from vertex. The shape formed is:
Reasoning: Distance from point $\ \Longrightarrow \ $ circle ; restricted to bisector gives intersection points.
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A point lies $4\ cm$ from $A$ and $4\ cm$ from line $l$. Which statement is true?
Reasoning: Depends on relative position of circle and line.
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