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Paragraph for Question 7 and 8
Let PQ be a focal chord of the parabolas $\mathrm{y}^{2}=4 \mathrm{ax} .$ The tangents to the parabola at $\mathrm{P}$ and $\mathrm{Q}$ meet at a point lying on the line $\mathrm{y}=2 \mathrm{x}+\mathrm{a}, \mathrm{a}>0 .$
Paragraph For Questions 11 and 12
Let a,r,s, t be nonzero real numbers. Let $P\left(a t^{2}, 2 a t\right), Q, R\left(a r^{2}, 2 a r\right)$ and $S\left(a s^{2}, 2 a s\right)$ be distinct points on the parabola $y^{2}=4 a x .$ Suppose that $P Q$ is the focal chord and lines $Q R$ and $P K$ are parallel, where $K$ is the point $(2 a, 0) .$
The value of $r$ is-
Q. The tangent PT and the normal PN to the parabola $y^{2}$ = 4ax at a point P on it meet its axis at points T and N, respectively. The locus of the centroid of the triangle PTN is a parabola whose
(A) vertex is $\left(\frac{2 \mathrm{a}}{3}, 0\right)$
(B) directrix is x = 0
(C) latus rectum is $\frac{2 \mathrm{a}}{3}$
(D) focus is (a, 0)
[JEE 2009, 4]
Ans. (A,D)
Q. Let A and B be two distinct point on the parabola $y^{2}$ = 4x. If the axis of the parabola touches a circle of radius r having AB as its diameter, then the slope of the line joining A and B can be –
(A) $\frac{-1}{\mathrm{r}}$
(B) $\frac{1}{\mathrm{r}}$
(C) $\frac{2}{\mathrm{r}}$
(D) $\frac{-2}{\mathrm{r}}$
[JEE 2010,3]
Ans. (C,D)
$\mathrm{t}_{1}+\mathrm{t}_{2}=\mathrm{r}$
$\frac{2}{\mathrm{r}}=\frac{2}{\mathrm{t}_{1}+\mathrm{t}_{2}}$
similarly $-\frac{2}{\mathrm{r}}$ is
also possible
Q. Consider the parabola $\mathrm{y}^{2}=8 \mathrm{x}$. Let $\Delta_{1}$ be the area of the triangle formed by the end points of its latus rectum and the point $\mathrm{P}\left(\frac{1}{2}, 2\right)$ on the parabola, and $\Delta_{2}$ be the area of the triangle formed by drawing tangents at $\mathrm{P}$ and at the end points of the latus rectum. Then $\frac{\Delta_{1}}{\Delta_{2}}$ is
[JEE 2011,4]
Ans. 2
Q. Let (x,y) be any point on the parabola $\mathrm{y}^{2}$ = 4x. Let P be the point that divides the line segment from (0,0) to (x,y) in the ratio 1 : 3. Then the locus of P is-
(A) $x^{2}=y$
(B) $\mathrm{y}^{2}=2 \mathrm{x}$
(C) $\mathrm{y}^{2}=\mathrm{x}$
(D) $x^{2}=2 y$
[JEE 2011,3]
Ans. (C)
Let P be (h, k)
on using section formula $\mathrm{P}\left(\frac{x}{4}, \frac{y}{4}\right)$
$\therefore \quad \mathrm{h}=\frac{x}{4}$ and $\mathrm{k}=\frac{y}{4}$
$\Rightarrow \quad \mathrm{x}=4 \mathrm{h}$ and $\mathrm{y}=4 \mathrm{k}$
$\because \quad(\mathrm{x}, \mathrm{y})$ lies on $\mathrm{y}^{2}=4 \mathrm{x}$
$\therefore \quad 16 \mathrm{k}^{2}=16 \mathrm{h} \quad \Rightarrow \mathrm{k}^{2}=\mathrm{h}$
Locus of point $\mathrm{P}$ is $\mathrm{y}^{2}=\mathrm{x}$
Q. Let L be a normal to the parabola $\mathrm{y}^{2}=4 \mathrm{x} .$ If $\mathrm{L}$ passes through the point $(9,6),$ then $\mathrm{L}$ is given by $-$
(A) y – x + 3 =0
(B) y + 3x – 33 = 0
(C) y + x – 15 = 0
(D) y – 2x + 12 = 0
[JEE 2011,3]
Ans. (A,B,D)
Equation of normal is $\mathrm{y}=\mathrm{mx}-2 \mathrm{m}-\mathrm{m}^{3}$
It passes through the point $(9,6)$ then
$6=9 \mathrm{m}-2 \mathrm{m}-\mathrm{m}^{3}$
$\Rightarrow \mathrm{m}^{3}-7 \mathrm{m}+6=0$
$\Rightarrow(\mathrm{m}-1)(\mathrm{m}-2)(\mathrm{m}+3)=0$
$\Rightarrow \quad \mathrm{m}=1,2,-3$
Equations of normals are
$\mathrm{y}-\mathrm{x}+3=0, \mathrm{y}+3 \mathrm{x}-33=0$
& $\quad \mathrm{y}-2 \mathrm{x}+12=0$
Q. Let S be the focus of the parabola $y^{2}=8 x \&$ let PQ be the common chord of the circle $x^{2}+y^{2}-2 x-4 y$ $=0$ and the given parabola. The area of the triangle PQS is
[JEE 2012, 4M]
Ans. 4
Q. If chord PQ subtends an angle $\theta$ at the vertex of $y^{2}=4 a x,$ then $\tan \theta=$
(A) $\frac{2}{3} \sqrt{7}$
(B) $\frac{-2}{3} \sqrt{7}$
(C) $\frac{2}{3} \sqrt{5}$
(D) $\frac{-2}{3} \sqrt{5}$
[JEE(Advanced) 2013, 3, (–1)]
Ans. (D)
Single tangent at the extrimities of a focal
chord will intersect on directrix.
$\therefore \quad \mathrm{M}\left(-\mathrm{a}, \mathrm{a}\left(\mathrm{t}_{1}+\mathrm{t}_{2}\right)\right)$
lies on $\mathrm{y}=2 \mathrm{x}+\mathrm{a}$
$\mathrm{a}\left(\mathrm{t}_{1}+\mathrm{t}_{2}\right)=-2 \mathrm{a}+\mathrm{a} \quad \Rightarrow \quad \mathrm{t}_{1}+\mathrm{t}_{2}=-1$
$\quad \& \quad \mathrm{t}_{1} \mathrm{t}_{2}=-1$
$\tan \theta=\left(\frac{\frac{2}{t_{1}}-\frac{2}{t_{2}}}{1+\frac{4}{t_{1} t_{2}}}\right)=\left(\frac{2\left(t_{2}-t_{1}\right)}{3}\right)$
$\because\left(t_{2}-t_{1}\right)=\left(t_{2}+t_{1}\right)^{2}-4 t_{1} t_{2}=5$
$t_{2}-t_{1}=\pm \sqrt{5}$
$\therefore \quad \tan \theta=\pm \frac{2 \sqrt{5}}{3}$
but $\theta$ is obtuse because $\mathrm{O}$ is the interior point of the circle for which $\mathrm{PQ}$ is diameter. $\therefore \quad \tan \theta=\frac{-2 \sqrt{5}}{3}$
chord will intersect on directrix.
$\therefore \quad \mathrm{M}\left(-\mathrm{a}, \mathrm{a}\left(\mathrm{t}_{1}+\mathrm{t}_{2}\right)\right)$
lies on $\mathrm{y}=2 \mathrm{x}+\mathrm{a}$
$\mathrm{a}\left(\mathrm{t}_{1}+\mathrm{t}_{2}\right)=-2 \mathrm{a}+\mathrm{a} \quad \Rightarrow \quad \mathrm{t}_{1}+\mathrm{t}_{2}=-1$
$\quad \& \quad \mathrm{t}_{1} \mathrm{t}_{2}=-1$
$\tan \theta=\left(\frac{\frac{2}{t_{1}}-\frac{2}{t_{2}}}{1+\frac{4}{t_{1} t_{2}}}\right)=\left(\frac{2\left(t_{2}-t_{1}\right)}{3}\right)$
$\because\left(t_{2}-t_{1}\right)=\left(t_{2}+t_{1}\right)^{2}-4 t_{1} t_{2}=5$
$t_{2}-t_{1}=\pm \sqrt{5}$
$\therefore \quad \tan \theta=\pm \frac{2 \sqrt{5}}{3}$
but $\theta$ is obtuse because $\mathrm{O}$ is the interior point of the circle for which $\mathrm{PQ}$ is diameter. $\therefore \quad \tan \theta=\frac{-2 \sqrt{5}}{3}$
Q. Length of chord PQ is
(A) 7a (B) 5a (C) 2a (D) 3a
[JEE(Advanced) 2013, 3, (–1)]
Ans. (B)
$\begin{aligned} \text { Length of focal chord } \\ \mathrm{PQ} &=\mathrm{a}\left(\mathrm{t}_{1}-\mathrm{t}_{2}\right)^{2} \\ &=\mathrm{a}\left[\left(\mathrm{t}_{1}+\mathrm{t}_{2}\right)^{2}-4 \mathrm{t}_{1} \mathrm{t}_{2}\right] \\ &=\mathrm{a}[1+4]=5 \mathrm{a} \end{aligned}$
Q. A line L: $y=m x+3$ meets $y-$ axis at $E(0,3)$ and the arc of the parabola $y^{2}=16 x, 0 \leq y$
$\leq 6$ at the point $F\left(x_{0}, y_{0}\right)$. The tangent to the parabola at $F\left(x_{0}, y_{0}\right)$ intersects the $y$ -axis at $G\left(0, y_{1}\right) .$
The slope $m$ of the line $L$ is chosen such that the area of the triangle EFG has a local maximum.
Match List-I with List-II and select the correct answer using the code given below the lists.
[JEE(Advanced) 2013, 3, (–1)]
[JEE(Advanced) 2013, 3, (–1)]
Ans. (A )
Q. The common tangents to the circle $\mathrm{x}^{2}+\mathrm{y}^{2}=2$ and the parabola $\mathrm{y}^{2}=8 \mathrm{x}$ touch the circle at the point $\mathrm{P}, \mathrm{Q}$ and the parabola at the points $\mathrm{R}, \mathrm{S}$. Then the area of the quadrilateral PQRS
is –
(A) 3 (B) 6 (C) 9 (D) 15
[JEE(Advanced)-2014, 3(–1)]
Ans. (D)
Q. The value of r is-
(A) $-\frac{1}{\mathfrak{t}}$
(B) $\frac{t^{2}+1}{t}$
(C) $\frac{1}{\mathrm{t}}$
(D) $\frac{\mathfrak{t}^{2}-1}{\mathfrak{t}}$
[JEE(Advanced)-2014, 3(–1)]
Ans. (D)
$\because \mathrm{PQ}$ is a focal chord
$\therefore$ co-ordinates of point $\mathrm{Q}$ are $=\left(\frac{\mathrm{a}}{\mathrm{t}^{2}},-\frac{2 \mathrm{a}}{\mathrm{t}}\right)$
$\mathrm{m}_{\mathrm{QR}}=\frac{2 \mathrm{a}\left(\mathrm{r}+\frac{1}{\mathrm{t}}\right)}{\mathrm{a}\left(\mathrm{r}^{2}-\frac{1}{\mathrm{t}^{2}}\right)}=\frac{2}{\left(\mathrm{r}-\frac{1}{\mathrm{t}}\right)}$
$\mathrm{m}_{\mathrm{PK}}=\frac{2 \mathrm{at}-0}{\mathrm{a}\left(\mathrm{t}^{2}-2\right)}=\frac{2 \mathrm{t}}{\mathrm{t}^{2}-2}$
Given $\mathrm{m}_{\mathrm{QR}}=\mathrm{m}_{\mathrm{PK}}$
$\Rightarrow \frac{2}{\mathrm{r}-\frac{1}{\mathrm{t}}}=\frac{2 \mathrm{t}}{\mathrm{t}^{2}-2} \Rightarrow \mathrm{r}=\frac{\mathrm{t}^{2}-2}{\mathrm{t}}+\frac{1}{\mathrm{t}}$
$\Rightarrow \mathrm{r}=\mathrm{t}-\frac{2}{\mathrm{t}}+\frac{1}{\mathrm{t}} \Rightarrow \mathrm{r}=\frac{\mathrm{t}^{2}-1}{\mathrm{t}}$
$\mathrm{m}_{\mathrm{QR}}=\frac{2 \mathrm{a}\left(\mathrm{r}+\frac{1}{\mathrm{t}}\right)}{\mathrm{a}\left(\mathrm{r}^{2}-\frac{1}{\mathrm{t}^{2}}\right)}=\frac{2}{\left(\mathrm{r}-\frac{1}{\mathrm{t}}\right)}$
$\mathrm{m}_{\mathrm{PK}}=\frac{2 \mathrm{at}-0}{\mathrm{a}\left(\mathrm{t}^{2}-2\right)}=\frac{2 \mathrm{t}}{\mathrm{t}^{2}-2}$
Given $\mathrm{m}_{\mathrm{QR}}=\mathrm{m}_{\mathrm{PK}}$
$\Rightarrow \frac{2}{\mathrm{r}-\frac{1}{\mathrm{t}}}=\frac{2 \mathrm{t}}{\mathrm{t}^{2}-2} \Rightarrow \mathrm{r}=\frac{\mathrm{t}^{2}-2}{\mathrm{t}}+\frac{1}{\mathrm{t}}$
$\Rightarrow \mathrm{r}=\mathrm{t}-\frac{2}{\mathrm{t}}+\frac{1}{\mathrm{t}} \Rightarrow \mathrm{r}=\frac{\mathrm{t}^{2}-1}{\mathrm{t}}$
Q. If st = 1, then the tangent at P and the normal at S to the parabola meet at a point whose ordinate is-
(A) $\frac{\left(\mathrm{t}^{2}+1\right)^{2}}{2 \mathrm{t}^{3}}$
(B) $\frac{\mathrm{a}\left(\mathrm{t}^{2}+1\right)^{2}}{2 \mathrm{t}^{3}}$
(C) $\frac{\mathrm{a}\left(\mathrm{t}^{2}+1\right)^{2}}{\mathrm{t}^{3}}$
(D) $\frac{\mathrm{a}\left(\mathrm{t}^{2}+2\right)^{2}}{\mathrm{t}^{3}}$
[JEE(Advanced)-2014, 3(–1)]
Ans. (B)
Q. If the normals of the parabola $y^{2}=4 x$ drawn at the end points of its latus rectum are tangents to the circle $(x-3)^{2}+(y+2)^{2}=r^{2},$ then the value of $r^{2}$ is
[JEE 2015, 4M, –0M]
Ans. 2
Q. Let the curve $C$ be the mirror image of the parabola $y^{2}=4 x$ with respect to the line $x+y+4=0 .$ If $A$ and $B$ are the points of intersection of $C$ with the line $y=-5$, then the distance between A and $B$ is
[JEE 2015, 4M, –0M]
Ans. 4
Let there be a point $\left(\mathrm{t}^{2}, 2 \mathrm{t}\right)$ on $\mathrm{y}^{2}=4 \mathrm{x}$
Clearly its reflection in $\mathrm{x}+\mathrm{y}+4=0$ is given by
$$
\frac{\mathrm{x}-\mathrm{t}^{2}}{1}=\frac{\mathrm{y}-2 \mathrm{t}}{1}=\frac{-2\left(\mathrm{t}^{2}+2 \mathrm{t}+4\right)}{2}
$$
$\therefore \quad \mathrm{x}=-(2 \mathrm{t}+4) \quad \& \mathrm{y}=-\left(\mathrm{t}^{2}+4\right)$
Now, $\mathrm{y}=-5 \quad \Rightarrow \quad \mathrm{t}=\pm 1$
$\therefore \quad \mathrm{x}=-6 \quad$ or $\quad \mathrm{x}=-2$
$\therefore \quad$ Distance between $\mathrm{A} \& \mathrm{B}=4$
Q. Let $P$ and $Q$ be distinct points on the parabola $y^{2}=2 x$ such that a circle with $P Q$ as diameter passes through the vertex $O$ of the parabola. If $P$ lies in the first quadrant and the area of the triangle $\Delta O P Q$ is $3 \sqrt{2},$ then which of the following is (are) the coordinates of $P ?$
(A) $(4,2 \sqrt{2})$
(B) $(9,3 \sqrt{2})$
(C) $\left(\frac{1}{4}, \frac{1}{\sqrt{2}}\right)$
(D) $(1, \sqrt{2})$
[JEE 2015, 4M, –2M]
Ans. (A,D)
Q. The circle $C_{1}: x^{2}+y^{2}=3,$ with centre at $O,$ intersects the parabola $x^{2}=2 y$ at the point $P$ in the first quadrant. Let the tangent to the circle $C_{1}$ at $P$ touches other two circles $C_{2}$ and $C_{3}$ at $R_{2}$ and $R_{3},$ respectively. Suppose $C_{2}$ and $C_{3}$ have equal radii $2 \sqrt{3}$ and centres $Q_{2}$ and $Q_{3}$ respectively. If $Q_{2}$ and $Q_{3}$ lie on the y-axis, then-
(A) $\mathrm{Q}_{2} \mathrm{Q}_{3}=12$
(B) $\mathrm{R}_{2} \mathrm{R}_{3}=4 \sqrt{6}$
(C) area of the triangle $\mathrm{OR}_{2} \mathrm{R}_{3}$ is $6 \sqrt{2}$
(D) area of the triangle $\mathrm{PQ}_{2} \mathrm{Q}_{3}$ is $4 \sqrt{2}$
[JEE (Advanced) 2016]
Ans. (A,B,C)
On solving $\mathrm{x}^{2}+\mathrm{y}^{2}=3$ and $\mathrm{x}^{2}=2 \mathrm{y}$ we get point $\mathrm{P}(\sqrt{2}, 1)$
Equation of tangent at $\mathrm{P}$
$\sqrt{2} \cdot \mathrm{x}+\mathrm{y}=3$
Let $\mathrm{Q}_{2}$ be $(0, \mathrm{k})$ and radius is $2 \sqrt{3}$
$\therefore\left|\frac{\sqrt{2}(0)+k-3}{\sqrt{2+1}}\right|=2 \sqrt{3}$
$\therefore \mathrm{k}=9,-3$
$\mathrm{Q}_{2}(0,9)$ and $\mathrm{Q}_{3}(0,-3)$
hence $\mathrm{Q}_{2} \mathrm{Q}_{3}=12$
$\mathrm{R}_{2} \mathrm{R}_{3}$ is internal common tangent of circle $\mathrm{C}_{2}$ and $\mathrm{C}_{3}$
$\begin{aligned} \therefore \mathrm{R}_{2} \mathrm{R}_{3} &=\sqrt{\left(\mathrm{Q}_{2} \mathrm{Q}_{3}\right)^{2}-(2 \sqrt{3}+2 \sqrt{3})^{2}} \\ &=\sqrt{12^{2}-48}=\sqrt{96}=4 \sqrt{6} \end{aligned}$
Perpendicular distance of origin $\mathrm{O}$ from $\mathrm{R}_{2} \mathrm{R}_{3}$ is equal to radius of circle $\mathrm{C}_{1}=\sqrt{3}$
Hence area of $\Delta \mathrm{OR}_{2} \mathrm{R}_{3}=\frac{1}{2} \times\left(\mathrm{R}_{2} \mathrm{R}_{3}\right) \sqrt{3}=\frac{1}{2} \cdot 4 \sqrt{6} \cdot \sqrt{3}=6 \sqrt{2}$
Perpendicular Distance of $\mathrm{P}$ from $\mathrm{Q}_{2} \mathrm{Q}_{3}=\sqrt{2}$
$\therefore$ Area of $\Delta \mathrm{PQ}_{2} \mathrm{Q}_{3}=\frac{1}{2} \times 12 \times \sqrt{2}=6 \sqrt{2}$
Q. Let $P$ be the point on the parabola $y^{2}=4 x$ which is at the shortest distance from the center Sof the circle $x^{2}+y^{2}-4 x-16 y+64=0$. Let $Q$ be the point on the circle dividing the line segment SP internally. Then-
(A) $\mathrm{SP}=2 \sqrt{5}$
(B) $\mathrm{SQ}: \mathrm{QP}=(\sqrt{5}+1): 2$
(C) the $\mathrm{x}$ -intercept of the normal to the parabola at $\mathrm{P}$ is 6
(D) the slope of the tangent to the circle at $\mathrm{Q}$ is $\frac{1}{2}$
[JEE (Advanced) 2016]
Ans. (A,C,D)
point $P$ lies on normal to parabola passing through centre of circle
$y+t x=2 t+t^{3}$
$8+2 t=2 t+t^{3}$
$t=2$
$t=2$
$P(4,4)$
$S P=\sqrt{(4-2)^{2}+(4-8)^{2}}$
$S P=2 \sqrt{5}$
$\mathrm{SQ}=2$
$\Rightarrow \mathrm{PQ}=2 \sqrt{5}-2$
$\frac{\mathrm{SQ}}{\mathrm{QP}}=\frac{1}{\sqrt{5}-1}=\frac{\sqrt{5}+1}{4}$
To find $\mathrm{x}$ intercept
put $\mathrm{y}=0 \mathrm{in}(\mathrm{i})$
$\Rightarrow \mathrm{x}=2+\mathrm{t}^{2}$
$\because=6$
$\because \quad$ Slope of common normal $=-\mathrm{t}=-2$
$\therefore \quad$ Slope of tangent $=\frac{1}{2}$
point $P$ lies on normal to parabola passing through centre of circle
$y+t x=2 t+t^{3}$
$8+2 t=2 t+t^{3}$
$t=2$
$t=2$
$P(4,4)$
$S P=\sqrt{(4-2)^{2}+(4-8)^{2}}$
$S P=2 \sqrt{5}$
$\mathrm{SQ}=2$
$\Rightarrow \mathrm{PQ}=2 \sqrt{5}-2$
$\frac{\mathrm{SQ}}{\mathrm{QP}}=\frac{1}{\sqrt{5}-1}=\frac{\sqrt{5}+1}{4}$
To find $\mathrm{x}$ intercept
put $\mathrm{y}=0 \mathrm{in}(\mathrm{i})$
$\Rightarrow \mathrm{x}=2+\mathrm{t}^{2}$
$\because=6$
$\because \quad$ Slope of common normal $=-\mathrm{t}=-2$
$\therefore \quad$ Slope of tangent $=\frac{1}{2}$
Q. If a chord, which is not a tangent, of the parabola $y^{2}=16 x$ has the equation $2 x+y=p,$ and midpoint $(h, k),$ then which of the following is (are) possible value(s) of $p, h$ and $k ?$
(A) p = 5, h = 4, k = –3
(B) p = –1, h = 1, k = –3
(C) p = –2, h = 2, k = –4
(D) p = 2, h = 3, k = –4
[JEE (Advanced) 2017]
Ans. (D)
Equation of chord with mid point $(\mathrm{h}, \mathrm{k}):$
$\mathrm{k} \cdot \mathrm{y}-16\left(\frac{\mathrm{x}+\mathrm{h}}{2}\right)=\mathrm{k}^{2}-16 \mathrm{h}$
$\Rightarrow 8 \mathrm{x}-\mathrm{ky}+\mathrm{k}^{2}-8 \mathrm{h}=0$
Comparing with $2 \mathrm{x}+\mathrm{y}-\mathrm{p}=0,$ we get
$\mathrm{k}=-4 ; 2 \mathrm{h}-\mathrm{p}=4$
only (D) satisfies above relation.
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