Question

Which of the following statement is(are) true?

• A. From any point on the directrix of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$, a pair of tangents are drawn to the auxiliary circle of the ellipse. The chord of contact will pass through the corresponding focus of the ellipse.
• B. If two tangents to a parabola $y^2=4ax$ intersect on the line $x=a$ then their chord of contact always passes through the foot of directrix.
• C. If $P(x,y)$ is such that it moves on a hyperbola $\left|\sqrt{(x-3{)}^2+(y-4{)}^2}-\sqrt{x^2+y^2}\right|=k^2+1$, then number of possible integral values of $k$ is equal to 3 .
• D. A circle passing through 3 co-normal points on the parabola $y^2=4x$ also passes through $(1,0)$.

Answer 1

Answer: (A), (B), (C)

(A) Let the point be $\left(\frac{a}{e},\beta \right)$.
Chord of contact is $T=0$
i.e., $x\left(\frac{a}{e}\right)+y\beta =a^2$, which is passing through ( $ae,0)$.
(B) $T\left(a{t}_1{t}_2,a\left(t_1+t_2\right)\right)\Rightarrow a{t}_1{t}_2=a\Rightarrow t_1{t}_2=1$
[Hence equation of PQ is $2x-\left(t_1+t_2\right)y+2a=0$ $(x+a)-\lambda y=0]$
$\Rightarrow$ chord PQ passes $(-a,0)$ which is foot of directrix $\therefore$ (B) is also true.
(C) $As\left|P{S}_1-P{S}_2\right|=$ constant so, for hyperbola
$0<k^2+1<5$
$\Rightarrow k^2<4$
$\Rightarrow -2<k<2$
So, number of integral values of $k$ are 3 (i.e., $k=-1,0,1$ )