Q1: The x-coordinate of a point P is twice its y-coordinate. If P is equidistant from Q (2, -5) and R(-3, 6), find the coordinates of P.
Ans:
Let the point P be (2k, k), Q(2,-5), R(-3, 6)
PQ = PR …Given
PQ^{2} = PR^{2} …[Squaring both sides
(2k – 2)^{2} + (k + 5)^{2} = (2k + 3)^{2} + (k – 6)^{2} …Given
4k^{2} + 4 – 8k + k^{2} + 10k + 25 = 4k^{2} + 9 + 12k + k^{2} – 12k + 36
⇒ 2k + 29 = 45
⇒ 2k = 45 – 29
⇒ 2k = 16
⇒ k = 8
Hence coordinates of point P are (16, 8).
Q2: Find the area of a parallelogram ABCD if three of its vertices are A(2, 4), B(2 + √3, 5) and C(2, 6).
Ans:
Since diagonal of a ||^{gm} divides it into two equal areas.
Area of ABCD (||^{gm}) = 2(Area of ∆ABC)
= 2√3 sq. units
Q3: Find the coordinates of a point P, which lies on the line segment joining the points A(-2, -2) and B(2, -4) such that AP = 3/7AB.
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Q4: If P(2, 4) is equidistant from Q(7, 0) and R(x, 9), find the values of x. Also find the distance P.
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PQ = PR …[Given]
PQ^{2} = PR^{2} … [Squaring both sides
∴ (7 – 2)^{2} + (0 – 4)^{2} = (x – 2)^{2} + (9 – 4)^{2}
⇒ 25 + 16 = (x – 2)^{2} + 25
⇒ 16 = (x – 2)^{2}
⇒ ±4 = x – 2 …[Taking sq. root of both sides
⇒ 2 ± 4 = x
⇒ x = 2 + 4 = 6 or x = 2 – 4 = -2
Q5: Find the value of k, if the points P(5, 4), Q(7, k) and R(9, – 2) are collinear.
Ans:
Given points are P(5, 4), Q(7, k) and R(9, -2).
x_{1} (y_{2} – y_{3}) + x_{2}(y_{3} – y_{1}) + x_{3}(y_{1} – y_{2}) = 0 …[∵ Points are collinear
∴ 5 (k + 2) + 7 (- 2 – 4) + 9 (4 – k) = 0
5k + 10 – 14 – 28 + 36 – 9k = 0
4 = 4k
∴ k = 1
Q6: If the points A(1, -2), B(2, 3), C(-3, 2) and D(-4, -3) are the vertices of parallelogram ABCD, then taking AB as the base, find the height of this parallelogram.
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Q7: The three vertices of a parallelogram ABCD are A(3, 4), B(-1, -3) and C(-6, 2). Find the coordinates of vertex D and find the area of ABCD.
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Q8: If (3, 3), (6, y), (x, 7) and (5, 6) are the vertices of a parallelogram taken in order, find the values of x and y.
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Let A (3, 3), B (6, y), C (x, 7) and D (5, 6).
Q9: Find the ratio in which the point P(x, 2) divides the line segment joining the points A(12, 5) and B(4, -3). Also, find the value of x.
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Q10: Point P(x, 4) lies on the line segment joining the points A(-5, 8) and B(4, -10). Find the ratio in which point P divides the line segment AB. Also find the value of x.
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