Find the conditions that the straight lines `y=m_1x+c_1, y=m_2x+c_2a n d\ y=m_3x+c_3` may meet in a point.

If the lines whose equations are y=m_1 x+ c_1 , y = m_2 x + c_2 and y=m_3 x + c_3 meet in a point, then prove that : m_1 (c_2 - c_3) + m_2 (c_3 - c_1) + m_3 (c_1 - c_2) =0

(i) Find the value of 'a' if the lines 3x-2y+8=0 , 2x+y+3=0 and ax+3y+11=0 are concurrent. (ii) If the lines y=m_(1)x+c_(1) , y=m_(2)x+c_(2) and y=m_(3)x+c_(3) meet at point then shown that : c_(1)(m_(2)-m_(3))+c_(2)(m_(3)-m_(1))+c_(3)(m_(1)-m_(2))=0

Find the condition that the straight line y = mx + c touches the hyperbola x^(2) - y^(2) = a^(2) .

The area of the triangle whose sides y=m_1 x + c_1 , y = m_2 x + c_2 and x=0 is

To find the condition that the three straight lines A_(1)x+B_(1)y+C_(1)=0, A_(2)x+B_(2)y+C_(2)=0 and A_(3)x+B_(3)y+C_(3)=0 are concurrent.

If three lines whose equations are y=m_1x+c_1,y=m_2x+c_2 and y=m_3x+c_3 are concurrent, then show that m_1(c_2-c_3)+m_2(c_3-c_1)+m_3(c_1-c_2)=0 .

Find the condition that the expressions a x^2-b x y+c y^2a n da_1x^2+b_1x y+c_1y^2 may have factors y-m xa n dm y-x , respectively.

Show that the area of the triangle formed buy the lines y=m_1x , y=m_2x\ a n d\ y=c is equal to (c^2)/4(sqrt(33)+sqrt(11))w h e r e\ m_1,\ m_2 are the roots of the equation x ^2+(sqrt(3)+2)x+sqrt(3)-1=0.

Show that the lines y-m_1 x-c_1 = 0 , y-m_2x-c_2 = 0 and y-m_3 x-c_3 =0 form an isosceles triangle with the first line as base if : (1+m_1 m_2) (m_1 - m_3) + (1+m_1 m_3) (m_1 - m_2) = 0

Find the point of intersection of the following pairs of lines: y=m_1x+a/(m_1)\ a n d\ y=m_2x+a/(m_2)dot