ParabolaHardBloom L3
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 $AC \cdot BC$ is equal to :
Options
A.2
B.1
C.$\frac{1}{2}$
D.$\frac{1}{4}$
Solution
Sol. [Key Concept: The foot of the perpendicular from the focus to any tangent lies on the tangent at the vertex; by power of the point, $(AC)(BC) = (CS)^{2}$.]
Using power of $C$
$$(BC)(AC) = (CS)^{2} = \left( \frac{1 + 2 - 1}{\sqrt{2}} \right)^{2} = 2$$
Create a free account to view solution
View Solution FreeMore Parabola Questions
The angle between normal chords of a parabola which subtend right angle at the vertex is tan-1 k, then k is...The tangent PT and the normal PN to the parabola y2 = 4ax at a point P on it meet its axis at points T and N, respective...Through a point $P( - 2,0)$, tangents $PQ$ and $PR$ are drawn to the parabola $y^{2} = 8x$. Two circles each passing thr...Length of the intercept on the normal at the point $P\left( at^{2},2at \right)$ of the parabola $y^{2} = 4ax$ made by th...Variable circle is described to pass through point $(1,0)$ and tangent to the curve $y = tan\left( \tan^{- 1}x \right)$....