Question

Find the locus of the points of intersection of two tangents to a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,$ if sum of their slopes is a constant $=\lambda$.

• A. $(\lambda x^2-a^2)=2xy$
• B. $\lambda (x^2-a^2)=2xy$
• C. $(x^2-\lambda a^2)=2xy$
• D. None of these

Answer 1

The correct option is B $\lambda (x^2-a^2)=2xy$

Any tangent to hyperbola is
$y=mx+\sqrt{a^2{m}^2-b^2}$
If it passes through the point $(h,k)$, then
${(k-mh)}^2=a^2{m}^2-b^2$
or $m^2(h^2-a^2)-2mhk+k^2+b^2=0$ ...(1)
Then there will be two tangents passing through $(h,k)$ whose slopes are given by (1)
Now $m_1+m_2=\lambda \therefore \frac{2hk}{h^2-a^2}=\lambda$
$\therefore$ Locus is $\lambda (x^2-a^2)=2xy$