Question

If the line x + y -1 = 0 is a tangent to a parabola with focus (1, 2) at A and intersects the directrix at B and tangent at vertex at C respectively, then AC · BC is equal to :

• A. 2
• B. 1
• C. 1/2
• D. -1/4

Answer 1

Answer: (A)

2

Answer 2

Let the tangent line be $L: x+y-1=0$ and the focus be $F=(1,2)$. A property of parabolas states that if a tangent line at point $A$ intersects the directrix at $B$ and the tangent at the vertex at $C$, then the product $AC \cdot BC$ is equal to the square of the distance from the focus $F$ to the tangent line $L$.

The distance $d$ from a point $(x_0, y_0)$ to a line $Ax + By + C = 0$ is given by the formula: $d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$

In this case, $(x_0, y_0) = (1, 2)$ and the line is $x + y - 1 = 0$ (so $A=1$, $B=1$, $C=-1$). The distance is: $d = \frac{|1(1) + 1(2) - 1|}{\sqrt{1^2 + 1^2}}$ $d = \frac{|1 + 2 - 1|}{\sqrt{1 + 1}}$ $d = \frac{|2|}{\sqrt{2}}$ $d = \sqrt{2}$

According to the property, $AC \cdot BC = d^2$. $AC \cdot BC = (\sqrt{2})^2 = 2$.