Coordinate Geometry – Assertion and Reason Questions

Q 1 :

Assertion (A): The value of y is 3, if the distance between the points P(2, -3) and Q(10, y) is 10.

Reason (R): Distance between two points is given by $\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A)
Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of Assertion (A)
Assertion (A) is true but reason (R) is false.
Assertion (A) is false but reason (R) is true.

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(1)

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A)

Q 2 :

Assertion (A): If the co-ordinates of the mid-points of the sides AB and AC of $ΔABC$ are D(3,5) and E(-3,-3) respectively, then BC = 20 units

Reason (R) : The line joining the mid points of two sides of a triangle is parallel to the third side and equal to half of it.

Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A)
Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A)
Assertion (A) is true but reason (R) is false.
Assertion (A) is false but reason (R) is true.

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(1)

For assertion : Distance between two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ $= \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

$∴$ Distance between $D E = \sqrt{{(- 3 - 3)}^{2} + {(- 3 - 5)}^{2}}$

$= \sqrt{36 + 64} = \sqrt{100} = 10 units$

By midpoint theorem, distance between $B C$ $= 2 \times distance between D E = 2 \times 10 = 20 units .$

So, assertion is true.

For reason: By midpoint theorem, the line joining the mid points of two sides of a triangle is parallel to the third side and equal to half of it. So, reason is also true.

Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A)

Q 3 :

Assertion (A): Point P(0,2) is the point of intersection of y-axis with the line 3x + 2y = 4.

Reason (R): The distance of point P(0,2) from the x-axis is 2 units.

Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A
A is true, but R is false.
A is false, but R is true.

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(2)

$T h e l i n e 3 x + 2 y = 4 c u t s t h e y - a x i s w h e n x = 0 . \Rightarrow 3 \times 0 + 2 y = 4 \Rightarrow 2 y = 4 \Rightarrow y = 2 ∴ P o i n t P (0, 2) i s t h e p o i n t o f i n t e r s e c t i o n o f y - a x i s w i t h t h e l i n e 3 x + 2 y = 4 .$

Also, the distance of any point P(x,y) from the x-axis is the magnitude of its y-coordinate.

$\Rightarrow Distance of the point P (0, 2) from x - axis is 2 units$

Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

Q 4 :

Assertion (A): The point on the x-axis which is equidistant from (2,-5) and (-2,9) is (8,0).

Reason (R): Points lying on the y-axis are always of the form (0,y).

Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A.
A is true, but R is false.
A is false, but R is true

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(4)

$P A = P B \Rightarrow \sqrt{(x - 2)^{2} + (0 + 5)^{2}} = \sqrt{(x + 2)^{2} + (0 - 9)^{2}}$

$S q u a r i n g b o t h s i d e s : (x - 2)^{2} + 25 = (x + 2)^{2} + 81$

$x^{2} - 4 x + 4 + 25 = x^{2} + 4 x + 4 + 81$

$- 8 x = 56 \Rightarrow x = \frac{56}{- 8} = - 7$

$T h u s, t h e p o i n t o n x - a x i s i s (- 7, 0), n o t (8, 0)$

$S o : A s s e r t i o n (A) i s f a l s e b u t R e a s o n (R) i s t r u e$

Q 5 :

Assertion (A): The point (–4, 6) divides the line segment joining A(–6, 10) and B(–7, 4) in the ratio 2 : 9.

Reason (R): The coordinates of the point P(x,y) which divides the line segment joining $A (x_{1}, y_{1}) a n d B (x_{2}, y_{2}) i n t e r n a l l y i n t h e r a t i o m_{1} : m_{2} a r e :$

$(\frac{m_{1} x_{2} + m_{2} x_{1}}{m_{1} + m_{2}}, \frac{m_{1} y_{2} + m_{2} y_{1}}{m_{1} + m_{2}})$

Both A and R are true, and R is the correct explanation of A
Both A and R are true, but R is not the correct explanation of A
A is true, but R is false.
A is false, but R is true

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(4)

Let point P(x,y) divide AB in the ratio 2 : 9.

Using section formula:

$x = \frac{2 (- 7) + 9 (- 6)}{2 + 9} = \frac{- 14 - 54}{11} = - \frac{68}{11} y = \frac{2 (4) + 9 (10)}{2 + 9} = \frac{8 + 90}{11} = \frac{98}{11} T h u s, C o o r d i n a t e s o f p o i n t P a r e (- \frac{68}{11}, \frac{98}{11}) . B u t t h e a s s e r t i o n c l a i m s P i s (- 4, 6), w h i c h i s i n c o r r e c t .$

Q 6 :

Assertion (A): If the points A(6, 1), B(8, 2), C(9, 4) and D(p, 3) are the vertices of a parallelogram (in order), then p = 7.

Reason (R): Diagonals of a parallelogram bisect each other, therefore midpoint of AC = midpoint of BD.

Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A
A is true, but R is false.
A is false, but R is true

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(1)

Given ABCD is a parallelogra, Diagonals bisect each other, so

$Midpoint of A C = Midpoint of B D (\frac{6 + 9}{2}, \frac{1 + 4}{2}) = (\frac{p + 8}{2}, \frac{3 + 2}{2}) S i m p l i f y i n g : (\frac{15}{2}, \frac{5}{2}) = (\frac{p + 8}{2}, \frac{5}{2}) C o m p a r i n g x - c o o r d i n a t e s : \frac{15}{2} = \frac{p + 8}{2} p + 8 = 15 \Rightarrow p = 7$

Topic Question Set

Q 1 :

Assertion (A): The value of y is 3, if the distance between the points P(2, -3) and Q(10, y) is 10.

Reason (R): Distance between two points is given by $\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

Q 2 :

Assertion (A): If the co-ordinates of the mid-points of the sides AB and AC of $ΔABC$ are D(3,5) and E(-3,-3) respectively, then BC = 20 units

Reason (R) : The line joining the mid points of two sides of a triangle is parallel to the third side and equal to half of it.

Q 3 :

Assertion (A): Point P(0,2) is the point of intersection of y-axis with the line 3x + 2y = 4.

Reason (R): The distance of point P(0,2) from the x-axis is 2 units.

Q 4 :

Assertion (A): The point on the x-axis which is equidistant from (2,-5) and (-2,9) is (8,0).

Reason (R): Points lying on the y-axis are always of the form (0,y).

Q 6 :

Assertion (A): If the points A(6, 1), B(8, 2), C(9, 4) and D(p, 3) are the vertices of a parallelogram (in order), then p = 7.

Reason (R): Diagonals of a parallelogram bisect each other, therefore midpoint of AC = midpoint of BD.

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