if the straight lines joining the origin to the points of intersection of the line 2x + y = c and the curve 3x²+ 4xy - 4x +1=0 are perpendicular, then c is :-

-1

-5/3

5/3

2

Question

if the straight lines joining the origin to the points of
intersection of the line 2x + y = c and the curve 3x
2 + 4xy - 4x +1=0 are
perpendicular, then c is :-

-1
-5/3
5/3
2

Aakash Sharma · Accepted Answer

Given : The straight lines joining the origin to the points of intersection of the line 2x + y = c or $\frac{2 x + y}{c} = 1$ and the curve 3x2 + 4xy - 4x + 1 = 0 are perpendicular So, the equation of the line joining the origin to the point of intersection of line and curve is $3 x^{2} + 4 x y - 4 x (\frac{2 x + y}{c}) + {(\frac{2 x + y}{c})}^{2} = 3 x^{2} + 4 x y - \frac{8 x^{2}}{c} - \frac{4 x y}{c} + \frac{4 x^{2}}{c^{2}} + \frac{y^{2}}{c^{2}} + \frac{4 x y}{c^{2}} = x^{2} (3 - \frac{8}{c} + \frac{4}{c^{2}}) + y^{2} (\frac{1}{c^{2}}) + 4 x y (1 - \frac{1}{c} + \frac{1}{c^{2}})$ Since angle between them is 90Â° So, Coefficient of x2 + Coefficient of y2 = 0 $\Rightarrow 3 - \frac{8}{c} + \frac{4}{c^{2}} + \frac{1}{c^{2}} = 0 \Rightarrow \frac{3 c^{2} - 8 c + 5}{c^{2}} = 0 \Rightarrow 3 c^{2} - 8 c + 5 = 0 \Rightarrow (3 c - 5) (c - 1) = 0 \Rightarrow c = \frac{5}{3}, 1$

if the straight lines joining the origin to the points of intersection of the line 2x + y = c and the curve 3x 2 + 4xy - 4x +1=0 are perpendicular, then c is :- -1 -5/3 5/3 2

if the straight lines joining the origin to the points of intersection of the line 2x + y = c and the curve 3x²+ 4xy - 4x +1=0 are perpendicular, then c is :-

-1

-5/3

5/3

2