Advertisements
Advertisements
प्रश्न
In the following, find the value of ‘a’ for which the given points are collinear
(a, 2 – 2a), (– a + 1, 2a) and (– 4 – a, 6 – 2a)
Advertisements
उत्तर
Let the points be A(a, 2 – 2a), B(– a + 1, 2a) C(– 4 – a, 6 – 2a).
Since the given points are collinear.
Area of a ∆ = 0
`1/2[(x_1y_2 + x_2y_3 + x_3y_1) - (x_2y_1 + x_3y_2 + x_1y_3)]` = 0
`1/2[((2"a"^2 + (- "a" + 1)(6 - 2"a") + (-4 - "a")(2 - 2"a"))), (-((- "a" + 1)(2 - 2"a") + 2"a"(-4 - "a") + (6 - 2"a")"a"))]` = 0
`1/2[((2"a"^2 - 6"a" + 2"a"^2 + 6 - 2"a" - 8 + 8"a" - 2"a" + 2"a"^2)-), ((-2"a" + 2"a"^2 + 2 - 2"a" - 8"a" - 2"a"^2 + 6"a" - 2"a"^2))]` = 0
`1/2[6"a"^2 - 10"a" + 8"a" - 2 - (2"a"^2 - 4"a"^2 - 12"a" + 6"a" + 2)]` = 0
6a2 – 2a – 2 – (– 2a2 – 6a + 2) = 0
6a2 – 2a – 2 + 2a2 + 6a – 2 = 0
8a2 + 4a – 4 = 0 ...(Divided by 4)
2a2 + a – 1 = 0
2a2 + 2a – a – 1 = 0
2a (a + 1) – 1 (a + 1) = 0
(a + 1) (2a – 1) = 0
a + 1 = 0 or 2a – 1 = 0
a = – 1 or 2a = 1 ⇒ a = `1/2`
The value of a = – 1 or `1/2`
APPEARS IN
संबंधित प्रश्न
Diagram of the adjacent picture frame has outer dimensions = 24 cm × 28 cm and inner dimensions 16 cm × 20 cm. Find the area of each section of the frame, if the width of each section is same.
The diagonal of a rectangular plot is 34 m and its perimeter is 92 m. Find its area.
Find the area of a quadrilateral one of whose diagonals is 30 cm long and the perpendiculars from the other two vertices are 19 cm and 11 cm respectively.
Calculate the area of quadrilateral ABCD, in which ∠ABD = 90°, triangle BCD is an equilateral triangle of side 24 cm and AD = 26 cm.
A wire when bent in the form of a square encloses an area of 484 m2. Find the largest area enclosed by the same wire when bent to from:
- An equilateral triangle.
- A rectangle of length 16 m.
The rate for a 1.20 m wide carpet is Rs. 40 per meter; find the cost of covering a hall 45 m long and 32 m wide with this carpet. Also, find the cost of carpeting the same hall if the carpet, 80 wide, is at Rs. 25. Per meter.
The shaded region of the given diagram represents the lawn in the form of a house. On the three sides of the lawn, there are flowerbeds having a uniform width of 2 m.
(i) Find the length and the breadth of the lawn.
(ii) Hence, or otherwise, find the area of the flower-beds.
Find the area and the perimeter of a square with diagonal 24 cm. [Take √2 = 1.41 ]
A quadrilateral field of unequal has a longer diagonal with 140m. The perpendiculars from opposite vertives upon this diagonal are 20m and 14m. Find the area of the field.
Find the area of quadrilateral PQRS.
