The points (a, b), (0, 0), (-a, -b) and (ab, b²) are

the vertices of a parallelogram

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UPSC NDA 2 2017 Mathematics Solved Paper

Question : 41 of 120

Marks: +1, -0

The points

(a, b)

(0, 0)

(-a, -b)

(ab, b^2)

are

Solution:

Concept:

Points are collinear if they all lie on the same straight line.

Explanation:

Check if all four points satisfy the line through

(0,0)

and

(a,b)

Slope

m = \frac{b-0}{a-0} = \frac{b}{a}

Equation of the line:

y = \frac{b}{a}x

ay = bx

Verify each point:

•

(a,b)

a(b)=b(a)

holds.

•

(0,0)

a(0)=b(0)

holds.

•

(-a,-b)

a(-b)=b(-a) \rightarrow -ab=-ab

holds.

•

(ab,b^2)

a(b^2)=b(ab) \rightarrow ab^2=ab^2

holds.

All points satisfy

ay=bx

, so they lie on the same line.

Answer:

The points are collinear.

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