The locus of point of intersection of the line y+m x= √a2 m2+b2 and m y-x=√a2+b2 m2 is

Question

The locus of point of intersection of the line y+m x= √a2 m2+b2 and m y-x=√a2+b2 m2 is (A) x2+y2=(1/a2)+(1/b2) (B) x2+y2=a2+b2 (C) x2-y2=a2-b2 (D) (1/x2)+(1/y2)=a2-b2

Tardigrade Admin · Accepted Answer

Let the point of intersection of given two lines is

P (h, k)

, which lies on both the lines.

∴ k + mh = a^{2} m^{2} + b^{2}

and

mk - h = a^{2} + b^{2} m^{2}

Squaring and adding, we get

(1 + m^{2}) k^{2} + (1 + m^{2}) h^{2} = a^{2} m^{2} + b^{2} + a^{2} + b^{2} m^{2}

∴

locus is

x^{2} + y^{2} = a^{2} + b^{2}

Q. The locus of point of intersection of the line $y + m x =$ $a^{2} m^{2} + b^{2}$ and $m y - x = a^{2} + b^{2} m^{2}$ is

$x^{2} + y^{2} = \frac{1}{a ^{2}} + \frac{1}{b ^{2}}$

$x^{2} + y^{2} = a^{2} + b^{2}$

$x^{2} - y^{2} = a^{2} - b^{2}$

$\frac{1}{x ^{2}} + \frac{1}{y ^{2}} = a^{2} - b^{2}$

Q. The locus of point of intersection of the line y+mx= a2m2+b2​ and my−x=a2+b2m2​ is

x2+y2=a21​+b21​

x2+y2=a2+b2

x2−y2=a2−b2

x21​+y21​=a2−b2

Let the point of intersection of given two lines is P(h,k), which lies on both the lines. ∴k+mh=a2m2+b2​ and mk−h=a2+b2m2​ Squaring and adding, we get (1+m2)k2+(1+m2)h2=a2m2+b2+a2+b2m2 ∴ locus is x2+y2=a2+b2

Q. The locus of point of intersection of the line $y + m x =$ $a^{2} m^{2} + b^{2}$ and $m y - x = a^{2} + b^{2} m^{2}$ is

$x^{2} + y^{2} = \frac{1}{a ^{2}} + \frac{1}{b ^{2}}$

$x^{2} + y^{2} = a^{2} + b^{2}$

$x^{2} - y^{2} = a^{2} - b^{2}$

$\frac{1}{x ^{2}} + \frac{1}{y ^{2}} = a^{2} - b^{2}$