The equation frac{x^2}{a} + frac{xy}{h} + fra | vedclass

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

Question

(B) The given equation is $\frac{x^2}{a} + \frac{xy}{h} + \frac{y^2}{b} = 0$. Multiplying by $abh$,we get $bhx^2 + abxy + ahy^2 = 0$. For the lines to

Vedclass · Accepted Answer

$4h^2 = ab$

Vedclass · Answer

(B) The given equation is $\frac{x^2}{a} + \frac{xy}{h} + \frac{y^2}{b} = 0$. Multiplying by $abh$,we get $bhx^2 + abxy + ahy^2 = 0$. For the lines to be coincident,the equation must be of the form $(px + qy)^2 = 0$,which is $p^2x^2 + 2pqxy + q^2y^2 = 0$. Comparing the coefficients of the two equations: $\frac{bh}{p^2} = \frac{ab}{2pq} = \frac{ah}{q^2} = k$ (constant). From $\frac{bh}{p^2} = \frac{ah}{q^2}$,we get $\frac{b}{p^2} = \frac{a}{q^2}$ $\Rightarrow \frac{q^2}{p^2} = \frac{a}{b}$ $\Rightarrow \frac{q}{p} = \sqrt{\frac{a}{b}}$. From $\frac{ab}{2pq} = \frac{bh}{p^2}$,we get $\frac{a}{2q} = \frac{h}{p} \Rightarrow \frac{p}{q} = \frac{2h}{a}$. Equating the ratios: $\sqrt{\frac{b}{a}} = \frac{2h}{a}$. Squaring both sides: $\frac{b}{a} = \frac{4h^2}{a^2}$ $\Rightarrow b = \frac{4h^2}{a}$ $\Rightarrow ab = 4h^2$. Thus,the condition for coincident lines is $4h^2 = ab$.

The equation $\frac{x^2}{a} + \frac{xy}{h} + \frac{y^2}{b} = 0$ $(a \neq 0, h \neq 0, b \neq 0)$ represents two coincident lines if:

Similar Questions

Let $PQR$ be a right-angled isosceles triangle,with the right angle at $Q(2, 1)$. If the equation of the line $PR$ is $2x + y = 3$,then the combined equation representing the pair of lines $PQ$ and $QR$ is:

If an angular bisector of the coordinate axes is one of the lines of $x^2+2ax y+3y^2=0$,then the sum of all possible values of $a$ is

The equation $8 x^2-24 x y+18 y^2-6 x+9 y-5=0$ represents a

If the equation $x^2 + y^2 + 2gx + 2fy + 1 = 0$ represents a pair of straight lines,then:

The joint equation of two lines passing through the origin,each making an angle of $30^{\circ}$ with the positive $Y$-axis,is

Vedclass Test Series

Exam Paper Generator

Online Exam Module