Question

The angle between the two tangents from the origin to the circle ${\left(x-7\right)}^2+{\left(y+1\right)}^2=25$

• A. $\frac{\pi }{4}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{2}$
• D. $\frac{\pi }{6}$

Answer 1

Answer: (C)

$\frac{\pi }{2}$

The equation of any line through the origin $(0,0)$

is $y=mx$

If it is a tangent to the circle $(x-7{)}^2+(y+1{)}^2=5^2$ then

$=>|\frac{7m+1}{{\sqrt{m}}^2+1}|=5$

$=>7m+1=25(m^2+1)$

$=>24m^2+14m-24=0-----------------\left(i\right)$

The equating being a quadratic in m,given two value of $m$, say $m_1$ and $m_2$.

These two value of m are the slope of the tangent drawn from the origin to the given circle from $\left(i\right)$ we have $m_1{m}_2=-1$

Hence the two tangent are perpendicular is $\frac{\pi }{2}$