If the pair of lines x^2-16pxy-y^2=0 and x^2- | vedclass

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

Question

(A) Let the given equations be:$x^2-16pxy-y^2=0 \quad (i)$$x^2-16qxy-y^2=0 \quad (ii)$The equation of the angle bisectors of a pair of lines $ax^2+2hx

Vedclass · Accepted Answer

$\frac{-1}{64}$

Vedclass · Answer

(A) Let the given equations be:$x^2-16pxy-y^2=0 \quad (i)$$x^2-16qxy-y^2=0 \quad (ii)$The equation of the angle bisectors of a pair of lines $ax^2+2hxy+by^2=0$ is given by $\frac{x^2-y^2}{a-b} = \frac{xy}{h}$.For equation $(i)$,$a=1, b=-1, h=-8p$. The bisectors are $\frac{x^2-y^2}{1-(-1)} = \frac{xy}{-8p} \implies \frac{x^2-y^2}{2} = \frac{xy}{-8p} \implies -8px^2+8py^2=2xy \implies -8px^2-2xy+8py^2=0 \quad (iii)$.For equation $(ii)$,$a=1, b=-1, h=-8q$. The bisectors are $\frac{x^2-y^2}{1-(-1)} = \frac{xy}{-8q} \implies \frac{x^2-y^2}{2} = \frac{xy}{-8q} \implies -8qx^2-2xy+8qy^2=0 \quad (iv)$.Since each pair bisects the angle between the other,equation $(i)$ must be the same as equation $(iv)$ and equation $(ii)$ must be the same as equation $(iii)$.Comparing $(i)$ and $(iv)$: $\frac{1}{-8q} = \frac{-16p}{-2} = \frac{-1}{8q}$. This leads to $1 = -64pq$,so $pq = \frac{-1}{64}$.

If the pair of lines $x^2-16pxy-y^2=0$ and $x^2-16qxy-y^2=0$ are such that each pair bisects the angle between the other pair,then $pq=$

Similar Questions

If the equation $2x^2 + kxy - 6y^2 + 3x + y + 1 = 0$ $(k > 0)$ represents a pair of straight lines,then their point of intersection is

If $(2,-1)$ is the point of intersection of the pair of lines $2x^2+axy+3y^2+bx+cy-3=0$,then $3a+2b+c=$

If the line $y=mx$ is one of the bisectors of $x^2+4xy-y^2=0$,then the value of $2m$ is:

If the equation $16x^2 - 24xy + 9y^2 - 8x + 6y - 35 = 0$ represents a pair of straight lines, then the equation of the locus of points equidistant from these two lines is...

If the lines represented by $x^2 - 2pxy - y^2 = 0$ are rotated about the origin by an angle $\theta$ in clockwise and counter-clockwise directions respectively,then the equation of the bisector of the angle between the lines in the new position is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module