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Parin Sir > Quantitative Aptitude (Maths) > Co–Ordinate Geometry > Examples 1 to 5 of Co–Ordinate Geometry > Examples 1 to 5 of Co–Ordinate Geometry

Examples 1 to 5 of Co–Ordinate Geometry

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  1. The distance between (a, 0) and (0, a) is …………..

Soln. Distance   =  \sqrt{(x_1-x_2)^2+(y_1-y_2)^2} =  \sqrt{(a-0)^2+(0-a)^2}  =  \sqrt2a

  1. By shifting the origin at (7, 2), new co–ordinates of (3, 4) are ………..

Soln. Here, Shifting the origin at (h, k) = (7, 2),  Old co–ordinates (x, y) = ( 3, 4)  &  new co–ordinates are (x’, y’ ), then

x’    =   x – h

 =   3 – 7  =  – 4

&      y’   =   y – k

=   4 – 2  =   2

 New co–ordinates are ( – 4, – 2 )

  1. Points (1, 3), (3, 1) and ( – 5, – 5) makes a ……………triangle.

Soln. Let, A(1, 3), B (3, 1) and C ( – 5, – 5 )

∴  By distance formula,  AB2 =  8,  BC2  =  100   and   AC2  =  100

∴  BC  =  AC   ⇒  Δ ABC  is an isosceles triangle.

  1. If (k, 1), (2, 1) and (3, 2) are collinear, then find the value of k.

Soln. Here, (k, 1), (2, 1) and (3, 2) are collinear.

∴  \left|\begin{matrix}k&1&1\\2&1&1\\3&2&1\\\end{matrix}\right|  =   0    ⇒   k (1 – 2) – 1 ( 2 – 3 ) + 1 ( 4 – 3 )  =  0   ⇒   k  =  2.

  1. Find the area of a triangle with vertices ( 12, 8 ), ( –2, 6 ) and ( 6, 0 ).

 Soln. Area of a triangle  = \frac{1}{2} |D|  ;  Where,  D  =  \left|\begin{matrix}x_1\ &y_1&1\\x_2&y_2&1\\x_3&y_3&1\\\end{matrix}\right|  = \left|\begin{matrix}12&8&1\\-2&6&1\\6&0&1\\\end{matrix}\right| =  100

∴  Area of  a triangle  =  \left |\frac{100}{2} \right | =   50.

 

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