Question

If the lines represented by ax? + 4xy + y² +8x+2 fy+c=0 intersect on y-axis, then (ſ. c)=​

Answer 1

Answer:

As s=ax2+2hxy+by2+2gx+2fy+c=0 represent a pair of line ∴∣∣∣∣∣∣∣∣​ahg​hbf​gfc​∣∣∣∣∣∣∣∣​=0

or abc+2fgh−af2−bg2−ch2=0....(1) Now say point ofintersection onY axis be (0,y1​ and  point of intersection of pair of line be obtained by solving the equations ∂x∂s​=0=∂y∂s​ ∴∂x∂s​=0⇒ax+by+g=0 ⇒⇒​{hy1​+g=0by1​+f=0​>(∗) and ∂y∂s​=0⇒bx+by+f=0  On compairing the equation given in (*) we get bg=fh and  bg2=fgh....(2) Again ax2+2hxy+by2+2gx+2fy+c=0 meet at y-axis ∴x=0 ⇒by2+2fy+c=0 whose roots must be equal ⇒by2+2fy+c=0 whose roots must be equal ∴f2=bcaf2=abc......(3) Now using (2) and (3) in equation (I) we have abc+2fgh−af2−bg2−ch2=0 ⇒(abc−af2)+(fgh−bg2)+fgh−ch2=0 ⇒0+0+fgh−ch2=0∴ch2=fgh.....(4) Now adding (2) and (4) 2fgh=ch2+bg2

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Answer 2

Answer:

by using formula y=mx+c