Question

If tangents are drawn on any focal chord of a parabola, then choose the correct option(s)

• A. both the tangents will always be perpendicular to each other
• B. extremities of the focal chord and intersection point of the tangents will lie on the circles discribed on the focal chord as diameter
• C. both the tangents need not be perpendicular to each other
• D. tangents will intersect each other at directrix

Answer 1

Answer: (A), (B), (D)

tangents will intersect each other at directrix

Let the end points of the focal chord are $(a{t}_1^2,2a{t}_1)$ and $(a{t}_2^2,2a{t}_2)$
Let slope of the tangents drawn at the ends of this focal chord are $m_1,m_2$
$\therefore m_1=\frac{1}{t_1},m_2=\frac{1}{t_2}$
From $m_1{m}_2=\frac{1}{t_1{t}_2}=-1[\because t_1,t_2\text{are endpoints of a focal chord}]$
Hence tangents will be perpendicular to each other and they intersect at directrix since $x$ coordinate of intersection point is $a{t}_1{t}_2=-a$.
Now from above image it is clear that extremities of focal chord and intersection point of tangents lies on circle ( as focal chord forms a right angled triangle with intersection point )and circle discribed on focal chord as diameter also touches the directrix.