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Parin Sir > Quantitative Aptitude (Maths) > Co–Ordinate Geometry > Examples 16 to 20 of Co–Ordinate Geometry > Examples 16 to 20 of Co–Ordinate Geometry

Examples 16 to 20 of Co–Ordinate Geometry

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  1. Find the perpendicular distance between two lines 3x + 4y – 2 = 0 and 6x + 8y – 1 = 0.

Soln. Here, both the lines are 6x + 8y – 4 = 0  and  6x + 8y – 1 = 0.

∴  p   =  \frac{\left|c_1-c_2\right|}{\sqrt{a^2+b^2}} =  \frac{\left|(-4)-(-1)\right|}{\sqrt{6^2+8^2}} = \frac{3}{10}

  1. If 3y + 4x = 1, y = x + 5 and 5y + bx = 3 are concurrent, then find the value of b.

Soln. Here, 4x + 3y – 1 = 0, x – y + 5 = 0 and bx + 5y – 3 = 0 are concurrent.

∴  \left|\begin{matrix}a_1&b_1&c_1\\a_3&b_2&c_2\\a_2&b_3&c_3\\\end{matrix}\right|= 0   ⇒  \left|\begin{matrix}4&3&-1\\1&-1&5\\b&5&-1\\\end{matrix}\right|   =  0   ⇒  b  =  6

  1. Find the equation of a line which makes X–intercept = 2 and passing through the intersection of x – 2y – 3 = 0 and 3x + 5y + 1 = 0.

Soln. The required line is :  x – 2y – 3  +  λ ( 3x + 5y + 1 ) =  0.

∴  (1 + 3λ)x + ( – 2 + 5λ) y + ( – 3 + λ)  =  0.

Now, X – intercept  =  - \frac{c}{a} \Rightarrow 2 = - \frac{-3+\lambda}{1+3\lambda} \Rightarrow \lambda = \frac{1}{7}

Hence, the equation of  a required line is :  10x – 9y – 20 = 0

  1. Is 4x2 + 4y2 – 12x + 24y + 29 =  0 represent a circle ? If  yes, find its centre and radius.

 Soln. First divide the whole equation by 4 we get :  x2 + y2 – 3x + 6y + \frac{29}{4}  =  0

Compare with,  x2 + y2 + 2gx + 2fy + c = 0, we get

g  =  - \frac{3}{2},  f  =  3  and  c  =  \frac{29}{4}

 Now, g2+ f2– c =  \frac{9}{4} + 9 - \frac{29}{4}

=    4  >  0

Now, centre  =  ( – g, – f)  =  \left(\frac{3}{2},\ -3\right)

Radius   =    \sqrt{g^2+f^2-c}  =  2

 ∴ The given eq. represents a circle.

  1. (4, – 5) lies ………….. the circle x2 + y2 – 4x + 6y – 5 = 0.

Soln. Here, (4) 2 + ( – 5) 2 – 4 (4) + 6 ( – 5) – 5   =  – 10  < 0

∴   (4, – 5)  lies  inside the given circle.

 

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