Question

The tangents to $x^2+y^2=25$ have slopes $\tan a$ and $\tan b$ intersects at $P$. If $|\cot a+\cot b|=2$ then

• A. Locus of $P$ will be a pair of hyperbolas
• B. Locus of $P$ will be a pair of ellipses
• C. Locus of $P$ pass through $(\pm 5,0)$
• D. Locus of $P$ pass through $(0,\pm 5)$

Answer 1

Answer: (A), (D)

The correct options are
A Locus of $P$ will be a pair of hyperbolas
D Locus of $P$ pass through $(0,\pm 5)$

Let the coordinates of the $P(h,k)$
Equation of the tangents from $P$ is
$y=mx\pm 5\sqrt{1+m^2}(k-mh{)}^2=25(1+m^2\left)m^2\right(h^2-25)-2mkh+k^2-25=0$
This is quadratic in $m$. Let the two roots be $m_1$ and $m_2$. Then ,
$m_1+m_2=\frac{2hk}{h^2-25}{m}_1{m}_2=\frac{k^2-25}{h^2-25}\because m_1=\tan a{m}_2=\tan b\Rightarrow \cot a+\cot b=\pm 2\frac{1}{m_1}+\frac{1}{m_2}=\pm 2\frac{2hk}{k^2-25}=\pm 2$
Locus of $P$ will be
$2xy=\pm 2(y^2-25)$