Exercises

1. A tangent PT is drawn at the point $\mathrm{P}(16,16)$ to the parabola ${\mathrm{y}}^2=16\mathrm{x}$. PT tangent intersect the $\mathrm{x}$-axis at $\mathrm{T}$. If $\mathrm{S}$ be the focus of the parabola, then $\mathrm{\angle }\mathrm{TPS}$ is equal to

(a). ${\tan }^{-1}\frac{1}{2}$

(b). $\frac{\pi }{4}$

(c). $\frac{1}{2}{\tan }^{-1}\frac{1}{2}$

(d). ${\tan }^{-1}\frac{3}{4}$

Show Answer

Solution: $\mathrm{ST}=4+\mathrm{AT}=16+4=20$

$\mathrm{PS}=4+16=20$

$\mathrm{\Delta }$ TPS is isosceles triangle

$\tan 2\theta =\frac{16-0}{16-4}=\frac{4}{3}=\frac{2\tan \theta }{1-{\tan }^2\theta }$

$2{\tan }^2\theta +3\tan \theta -2=0$

$(2\tan \theta -1)(\tan \theta +2)=0$

$\tan \theta =\frac{1}{2},\tan \theta =-2$ (Not possible)

$\theta ={\tan }^{-1}\frac{1}{2}$ ( $\theta$ is acute angle)

Answer: a

2. If a, b, $c$ are distinct positive real numbers such that the parabolas $y^2=4ax$ and $y^2$ $=4\mathrm{c}(\mathrm{x}-\mathrm{b})$ will have a common normal, then

(a). $0<\frac{\mathrm{b}}{\mathrm{a}-\mathrm{c}}<1$

(b). $\frac{b}{a-c}<0$

(c). $1<\frac{\mathrm{b}}{\mathrm{a}-\mathrm{c}}<2$

(d). $\frac{b}{a-c}>2$

3. If $\mathrm{AB}$ be a chord of the parabola ${\mathrm{y}}^2=4\mathrm{ax}$ with vertex at $\mathrm{A}$. $\mathrm{BC}$ is perpendicular to $\mathrm{AB}$ such that it meets the axis at $\mathrm{C}$. The projection of the $\mathrm{BC}$ on the axis of parabola is

(a). $2\mathrm{a}$

(b). $4\mathrm{a}$

(c). $8\mathrm{a}$

(d). $16\mathrm{a}$

Show Answer

Solution: Let coordinates of B be ( $\mathrm{x},\mathrm{y})$

In $\mathrm{△}\mathrm{ABD},\tan \theta =\frac{\mathrm{BD}}{\mathrm{AD}}=\frac{\mathrm{y}}{\mathrm{x}}$

In $\mathrm{△}\mathrm{BCD},\tan (90-\theta )=\frac{\mathrm{BD}}{\mathrm{DC}}$

$\therefore \mathrm{DC}=\mathrm{y}\cdot \frac{\mathrm{y}}{\mathrm{x}}=\frac{4\mathrm{ax}}{\mathrm{x}}=4\mathrm{a}$

Answer: b

4. A circle is descirbed whose centre is the vertex and whose diameter is three-quarters of the latus rectum of the parabola ${\mathrm{y}}^2=4\mathrm{ax}$. If $\mathrm{PQ}$ is the common-chord of the circle and the parabola and ${\mathrm{L}}_1{\mathrm{L}}_2$ is the latus rectum, then the area of the trapezium ${\mathrm{PL}}_1{\mathrm{L}}_2\mathrm{Q}$ is

(a). $\frac{2+\sqrt{2}}{2}{a}^2$

(b). $4a^2$

(c). $2{\sqrt{2}}^2$

(d). $3\sqrt{2}{\mathrm{a}}^2$

Show Answer

Solution: Centre of circle $(0,0)$

diameter $=\frac{3}{4}\cdot 4\mathrm{a}=3\mathrm{a}$

${\mathrm{Eq}}^{\mathrm{n}}$ of circle ${\mathrm{x}}^2+{\mathrm{y}}^2=\frac{9{\mathrm{a}}^2}{4}\dots ..(1)$

${\mathrm{Eq}}^{\mathrm{n}}$ of parabola ${\mathrm{y}}^2=4\mathrm{ax}\dots \dots (2)$

coordinates of $\mathrm{P}$ and $\mathrm{Q}$, we get after solving (1) and (2).

$\begin{array}{rl}{x}^2+4ax=\frac{9a^2}{4}& \\ (x+2a{)}^2={\frac{5a}{2}}^{2}x& =-2a\pm \frac{5a}{2}\\ & x=\frac{a}{2},\frac{-9a}{2}(\text{ not possible })\end{array}$

$P\frac{a}{2},\sqrt{2}a,Q\frac{a}{2},-\sqrt{2}ay=\pm \sqrt{2}a$

$\mathrm{PQ}=2\sqrt{2}\mathrm{a},{\mathrm{L}}_1{\mathrm{L}}_2=4\mathrm{a}$

Area of trapezium $=\frac{1}{2}(\mathrm{PQ}+{\mathrm{L}}_1{\mathrm{L}}_2)\mathrm{x}$ distance between them.

$\begin{array}{rl}& =\frac{1}{2}(2\sqrt{2}a+4a)\times a-\frac{a}{2}\\ & =\frac{(\sqrt{2}+2)}{2}{a}^2\end{array}$

Answer: a

5. From the point $(15,12)$ three normals ae drawn to the parabola $y^2=4x$, then centroid of triangle formed by three co-normal points is

(a). $(5,0)$

(b). $(5,4)$

(c). $(9,0)$

(d). $\frac{26}{3},0$

6. Aray of light travels along a line $\mathrm{y}=4$ and strikes the surface of a curve $y^2=4(x+y)$ then equation of the line along reflected ray travel is

(a). $x+1=0$

(b). $\mathrm{y}-2=0$

(c). $\mathrm{x}=0$

(d). $\mathrm{x}-2=0$

Show Answer

Solution: $y^2-4y=4x$

$(\mathrm{y}-2{)}^2=4(\mathrm{x}+1)$

Focus $(0,2)$

Incident ray is parallel to axis of the parabola, so reflected ray passes through focus $(0,2)$ i.e. $x=0$

Answer: c

7. Let $\mathrm{P}$ be a point of the parabola ${\mathrm{y}}^2=3(2\mathrm{x}-3)$ and $\mathrm{M}$ is the foot of perpendicular drawn from

Pon the directrix of the parabola, then length of each side of an equilateral triangle SMP, where S is focus of the parabola is

(a). 6

(b). 8

(c). 10

(d). 11

Show Answer

Solution: Equation of parabola is

${\mathrm{y}}^2=6\mathrm{x}-\frac{3}{2}$

focus $S(3,0)$

equation of directrix $x=0$

P $\frac{3}{2}+\frac{3}{2}{\mathrm{t}}^2,3\mathrm{t}$

Coordinates of $\mathrm{M}(0,3\mathrm{t})$

$\mathrm{MS}=\sqrt{9+9{\mathrm{t}}^2}$

$\mathrm{MP}=\frac{3}{2}+\frac{3}{2}{\mathrm{t}}^2$, But MS $=\mathrm{MP}$

$9+9{\mathrm{t}}^2=\frac{9}{4}+\frac{9}{4}{\mathrm{t}}^2+\frac{9}{4}{\mathrm{t}}^2$

${\mathrm{t}}^2=3$

Length of side $=6$

Answer: a

Practice questions

1. The point $(2a,a)$ lies inside the region bounded by the parabola $x^2=4y$ and its latus rectum. Then

(a). $0<\mathrm{a}\le 1$

(b). $0<\mathrm{a}<1$

(c). $0\le \mathrm{a}\le 1$

(d). $\mathrm{a}<1$

2. Two perpendicular tangents to $y^2=4ax$ always intersect on the line

(a). $x+a=0$

(b). $\mathrm{x}-\mathrm{a}=0$

(c). $x+2a=0$

(d). $y+2a=0$

3. ${\mathrm{C}}_1:{\mathrm{y}}^2=8\mathrm{x}$ and ${\mathrm{C}}_2:{\mathrm{x}}^2+{\mathrm{y}}^2=2$. Then

(a). ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ have only two common tangents which are mutually perpendicular

(b). ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ have two common tangents which are parallel to each other.

(c). does not have any common tangent.

(d). ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ have four common tangents.

4. Two common tangents to the circle $x^2+y^2=2a^2$ and $y^2=8ax$ are

(a). $y=(x+a)$

(b). $x=\pm (y+2a)$

(c). $y=\pm (x+2a)$

(d). $x=\pm (y+a)$

5. The number of points with integral coordinates that lie in the interior of the circle $x^2+y^2=16$ and the parabola ${\mathrm{y}}^2=4\mathrm{x}$ are

(a). 6

(b). 8

(c). 10

(d). 12

6. The vertex of the parabola $x^2+y^2-2xy-4x+4=0$ is at

(a). $(-\frac{1}{2},-\frac{1}{2})$

(b). $(-1,-1)$

(c). $(1,1)$

(d). $(+\frac{1}{2},\frac{1}{2})$

7. The length of the latus rectum of the parabola $2\{(x-a{)}^2+(y-a{)}^2\}=(x+y{)}^2$ is

(a). $\sqrt{2}$ a

(b). $2\mathrm{a}$

(c). $2\sqrt{2}$ a

(d). $3\sqrt{2}\mathrm{a}$

8. The point on $y^2=4ax$ nearest to the focus is

(a). $(0,0)$

(b). $(\mathrm{a},2\mathrm{a})$

(c). $(a,-2a)$

(d). $(\frac{a}{4},a)$

9. The angle between the tangents drawn from the origin to the parabola $y^2=4a(x-a)$ is

(a). ${45}^{\circ }$

(b). ${60}^{\circ }$

(c). ${90}^{\circ }$

(d). ${\tan }^{-1}\frac{1}{2}$

10. The circle $x^2+y^2+2\lambda x=0,\lambda \epsilon R$, touches the parabola $y^2=4x$ externally. Then

(a). $\lambda >0$

(b). $\lambda <0$

(c). $\lambda >1$

(d). none of these