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Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.
Assertion (A): The ordinate of a point A on yaxis is 5 and B has coordinates (–3, 1). Then the length of AB is 5 units.
Reason (R): The point A(2, 7) lies on the perpendicular bisector of line segment joining the points P(6, 5) and Q(0, –4).
Here, A → (0 , 5) and B → (– 3 , 1)
= 5 units
∴ Assertion is correct.
In case of reason: If A (2, 7) lies on perpendicular bisector of P (6, 5) and Q (0, –4), then
AP = AQ
∴ By using Distance Formula,
As, AP ≠ AQ
Therefore, A does not lie on the perpendicular bisector of PQ.
∴ Reason is incorrect.
Hence, assertion is correct but reason is incorrect.
Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.
Assertion : The area of the triangle with vertices (5 , 1), (3, 5), (5, 2), is 32 square units.Reason : The point (x, y) divides the line segment joining the points (x_{1}, y_{1}) and (x_{2}, y_{2}) in the ratio k : 1 externally then
and section formula (externally), we have
Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.
Assertion : The coordinates of the point which divides the join of A(5, 11) and B(4,7) in the ratio 7 : 2 is (2, 3) Reason : The coordinates of the point P(x, y) which divides the line segment joining the points A(x_{1}, y_{1}) and B(x_{2}, y_{2}) in the ratio m_{1} : m_{2} is
So, Reason is correct.
Here, x_{1} = 5, y_{1} = 11, x_{2} = 4, y_{2} = 7, m_{1} = 7, m_{2} = 2
So, Assertion is also correct
Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.
Assertion (A): △ABC with vertices A(–2, 0), B(2, 0) and C(0, 2) is similar to △DEF with vertices D(–4, 0), E(4, 0) and F(0, 4).
Reason (R): A circle has its centre at the origin and a point P(5, 0) lies on it. The point Q(6, 8) lies outside the circle.
∴ Distance between A (2, 0) and B (2, 0),
[∵ distance between the points (x_{1}, y_{1}) and (x_{2}, y_{2})
Similarly, distance between B (2, 0) and C (0, 2),
In △ABC, distance between C (0, 2) and A (2, 0),
Distance between F (0, 4) and D (–4, 0),
Distance between F (0, 4) and E (4, 0),
and distance between E (4, 0) and D (–4, 0),
Here, we see that sides of DABC and DFDE are proportional.
Therefore, by SSS similarity rule, both the triangles are similar.
∴ Assertion is correct.
In case of reason: Point Q (6, 8) will lie outside the circle if its distance from the centre of circle is greater than the radius of the circle.
Distance between centre O (0, 0) and P (5, 0),
As, point P lies at the circle, therefore, OP = Radius of circle.
Distance between centre O (0, 0) and Q (6, 8),
As OQ > OP, therefore, point Q (6, 8) lies outside of the circle.
∴ Reason is correct.
Hence, both assertion and reason are correct but reason is not the correct explanation for assertion.
Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.
Assertion : The points (k, 2 2k), ( k+ 1,2k) and ( 4  k, 6  2k) are collinear if k = 1/2.
Reason : Three points A,B and C are collinear in same straight line, if AB + BC = AC.
(−k + 1, 2k), (k, 2 − 2k),(−4 −k, 6 − 2k)
Since, these points are collinear. Therefore the area of triangle formed by the triangle formed by the points will be zero.
Therefore,
(−k + 1)(2 − 2k − 6 + 2k) + k(6 − 2k − 2k) + (−4 − k)(2k − 2 + 2k) = 0
(−k + 1)(−4) + k(6 − 4k) + (−4 − k)(4k − 2) = 0
4k − 4 + 6k − 4k^{2} − 16k + 8 − 4k^{2 }+ 2k = 0
−2k^{2} − k + 1 = 0
2k^{2} + k − 1 = 0
2k^{2} + 2k − k − 1 = 0
2k(k + 1) − 1(k + 1) = 0
(k + 1)(2k − 1) = 0
k = −1 or k = 1/2.
Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.
Assertion : If the points A(4, 3) and B(x, 5) lies on a circle with the centre O(2,3) then the value of x is 2.
Reason : The midpoint of the line segment joining the points P(x_{1}, y_{1}) and Q(x_{2}, y_{2}) is
So, Reason is correct.
Given, the points A (4,3) and B (x, 5) lie on a circle with center O(2,3).
Then OA = OB ⇒(OA)^{2} = (OB)^{2}
⇒ (4 – 2)^{2} + (3 – 3)^{2} = (x – 2)^{2} + (5 – 3)^{2}
⇒ (2)^{2} +(0)^{2} = (x – 2)^{2} + (2)^{2} ⇒ 4 = (x – 2)^{2} + 4 ⇒(x – 2)^{2} = 0
⇒ x – 2 = 0 ⇒ x = 2
So, Assertion is correct
Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.
Assertion (A): If the distance between the point (4, p) and (1, 0) is 5, then the value of p is 4.
Reason (R): The point which divides the line segment joining the points (7, – 6) and (3, 4) in ratio 1 : 2 internally lies in the fourth quadrant.
where, (x_{1}, y_{1}) = (4, p)
(x_{2}, y_{2}) = (1, 0)
And, d = 5
Put the values, we have
5^{2} = (1 − 4)^{2} + (0 – p)^{2}
25 = (–3)^{2} + (–p)^{2}
25 – 9 = p^{2}
16 = p^{2}
+4, –4 = p
∴ Assertion is incorrect.
In case of reason:
Let (x, y) be the point
Here, x_{1} = 7, y_{1} = –6, x_{2} = 3, y_{2} = 4, m = 1 and n = 2
So, the required point lies in IV^{th} quadrant.
∴ Reason is correct.
Hence, assertion is incorrect but reason is correct.
Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.
Assertion : Centroid of a triangle formed by the points (a, b) (b, c), and (c, a) is at origin, Then a + b + c = 0 .
Reason : Centroid of a △ABC with vertices A (x_{1}, y_{1}), B(x_{2}, y_{2}) and C (x_{3}, y_{3}) is given by
a + b + c = 0
Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.
Assertion : The possible values of x for which the distance between the points A(x, 1) and B(5, 3) is 5 units are 2 and 8.
Reason : Distance between two given points A(x_{1}, y_{1}) and B(x_{2}, y_{2}) is given by,
So, Reason is correct.
Now, AB = 5 ⇒ AB^{2} = 25
⇒ (x – 5)^{2} + (1  3)^{2} = 25
⇒ (x – 5)^{2} = 25 – 16 = 9
⇒ x – 5 = ±3
⇒ x = 5 ± 3
⇒ x = 2, 8
So, Assertion is also correct
Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.
Assertion : Midpoint of a line segment divides line in the ratio 1 : 1.
Reason : If area of triangle is zero that means points are collinear.
Let(x_{1}, y_{1}, z_{1}) = (0, 0, 0)
(x, y ,z) = (2, −3, 3)
(x_{2}, y_{2}, z_{2}) = (−2, 3, −3)
By section formula:
x = m_{1} + m_{2} m_{1} x 2 + m_{2} x 1
⇒ 2 = k + 1 k.(−2) + 1.(0)
⇒ k = 4 − 1 on neglecting the negative sign
we get, therefore ratio is 1:4.
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