If one of the lines represented by the equati | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The equations of the given lines are:$ax^2 + 2hxy + by^2 = 0$ ..... $(i)$$a'x^2 + 2h'xy + b'y^2 = 0$ ..... $(ii)$Let the common line be $y = mx$.

Vedclass · Accepted Answer

$(ab' - a'b)^2 = 4(ah' - a'h)(hb' - h'b)$

Vedclass · Answer

(A) The equations of the given lines are:$ax^2 + 2hxy + by^2 = 0$ ..... $(i)$$a'x^2 + 2h'xy + b'y^2 = 0$ ..... $(ii)$Let the common line be $y = mx$. Substituting this into both equations:$a + 2hm + bm^2 = 0$ ..... $(iii)$$a' + 2h'm + b'm^2 = 0$ ..... $(iv)$Using the method of cross-multiplication to eliminate $m$:$\frac{m^2}{2ha' - 2h'a} = \frac{-m}{ab' - a'b} = \frac{1}{2bh' - 2b'h}$From the first and third terms: $m^2 = \frac{2(ha' - h'a)}{2(bh' - b'h)} = \frac{ha' - h'a}{bh' - b'h}$From the second and third terms: $m = -\frac{ab' - a'b}{2(bh' - b'h)}$Squaring the expression for $m$: $m^2 = \frac{(ab' - a'b)^2}{4(bh' - b'h)^2}$Equating the two expressions for $m^2$:$\frac{ha' - h'a}{bh' - b'h} = \frac{(ab' - a'b)^2}{4(bh' - b'h)^2}$$(ab' - a'b)^2 = 4(ha' - h'a)(bh' - b'h) = 4(ah' - a'h)(hb' - h'b)$

If one of the lines represented by the equation $ax^2 + 2hxy + by^2 = 0$ is coincident with one of the lines represented by $a'x^2 + 2h'xy + b'y^2 = 0$,then:

Similar Questions

If the slopes of the lines represented by the equation $6x^2 + 2hxy + 4y^2 = 0$ are in the ratio $2:3$,then the value of $h$ such that both the lines make acute angles with the positive $X$-axis measured in the positive direction is

If the slope of one of the lines represented by the equation $ax^2 + 2hxy + by^2 = 0$ is $\lambda$ times that of the other,then:

The slopes of the lines represented by $6x^2 + 2hxy + y^2 = 0$ are in the ratio $2:3$,then $h =$

If the slope of one of the two lines represented by $\frac{x^2}{a} + \frac{2xy}{h} + \frac{y^2}{b} = 0$ is twice that of the other,then $ab : h^2 = $

If $4x^2-5xy+y^2=0$ represents a pair of lines with slopes $m_1$ and $m_2$,then the value of $|m_1-m_2|$ equals

Vedclass Test Series

Exam Paper Generator

Online Exam Module