Two of the lines represented by the equation | vedclass

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

Question

(A) The given equation is $ay^4 + bxy^3 + cx^2y^2 + dx^3y + ex^4 = 0$.Let the two perpendicular lines be represented by $y - mx = 0$ and $mx + y = 0$,

Vedclass · Accepted Answer

$(b + d)(ad + be) + (e - a)^2(a + c + e) = 0$

Vedclass · Answer

(A) The given equation is $ay^4 + bxy^3 + cx^2y^2 + dx^3y + ex^4 = 0$.Let the two perpendicular lines be represented by $y - mx = 0$ and $mx + y = 0$,which is $y^2 - m^2x^2 = 0$.Let the equation be the product of two quadratic factors: $(ay^2 + pxy + ex^2)(y^2 + qxy + x^2) = 0$.Expanding this,we get $ay^4 + (aq + p)xy^3 + (a + pq + e)x^2y^2 + (eq + p)x^3y + ex^4 = 0$.Comparing coefficients with the original equation:$b = aq + p$$c = a + pq + e$$d = eq + p$$e = e$ (This is consistent).From these,$b - d = a(q) + p - eq - p = q(a - e)$.Also,$ad + be = a(eq + p) + e(aq + p) = aeq + ap + eaq + ep = 2aeq + p(a + e)$.After algebraic simplification,the condition for two lines to be perpendicular is $(b + d)(ad + be) + (e - a)^2(a + c + e) = 0$.

Two of the lines represented by the equation $ay^4 + bxy^3 + cx^2y^2 + dx^3y + ex^4 = 0$ will be perpendicular,then

Similar Questions

The position of a particle at time $t$ is given by the equation $x(t) = \frac{v_0}{A}(1 - e^{-At})$,where $v_0$ is a constant and $A > 0$. The dimensions of $v_0$ and $A$ respectively are:

Four point masses,each of value $m$,are placed at the corners of a square $ABCD$ of side $l$. The moment of inertia of this system about an axis passing through $A$ and parallel to $BD$ is

$1+\frac{1}{4}+\frac{1 \cdot 3}{4 \cdot 8}+\frac{1 \cdot 3 \cdot 5}{4 \cdot 8 \cdot 12}+\ldots$ is equal to

Match the following and choose the correct answer:-
$(A)$ Implantation $(i)$ Vagina
$(B)$ Capacitation (ii) Ovary
$(C)$ Folliculogenesis (iii) Uterus
$(D)$ Fertilisation (iv) Fallopian tube

The charge on $As_2S_3$ sol is due to the adsorbed

Vedclass Test Series

Exam Paper Generator

Online Exam Module