Examples

1. If the chord of contact of tangent from a point $\mathrm{P}$ to the parabola $\mathrm{y}^{2}=4 \mathrm{ax}$ touches the parabola $\mathrm{x}^{2}=4$ by. The locus of $\mathrm{P}$ is

(a) Parabola

(b) Hyperbola

(c) ellipse

(d) Circle

2. Let $\mathrm{P}$ and $\mathrm{Q}$ be points $(4,-4)$ and $(9,6)$ on the parabola $\mathrm{y}^{2}=4 \mathrm{a}(\mathrm{x}-\mathrm{b})$. $\mathrm{R}$ is a point on the parabola so that area is maximum, then

(a) $\angle \mathrm{PRQ}=90^{\circ}$

(b) the point $\mathrm{R}$ is $(4,4)$

(c) the point $\mathrm{R}$ is $\frac{1}{4}, 1$

(d) None of these

Show Answer

Solution: $(4,-4)$ lies on $y^{2}=4 a(x-b) \quad 16=4 a(4-b)$

$(9,6)$ lies on $\mathrm{y}^{2}=4 \mathrm{a}(\mathrm{x}-\mathrm{b}) \quad 36=4 \mathrm{a}(9-\mathrm{b})$

$ \begin{aligned} & \frac{16}{36}=\frac{4-b}{9-b} \quad b=0 \\ & y^{2}=4 x \end{aligned} $

Let $R$ be $\left(t^{2}, 2 t\right)$ on the Parabola thus

$ \begin{aligned} & \text { area } \quad=\frac{1}{2}\left|\begin{array}{ccc} 4& -4 & 1 \\ 9& 6 & 1 \\ t^{2} & 2 t & 1 \end{array}\right|=\frac{1}{2}\left|\left\{4(6-2 t)+4\left(9-t^{2}\right)+18 t-6 t^{2}\right\}\right| \\ & =\quad \frac{1}{2}\left|\left(24-8 t+36-4 t^{2}+18 t-6 t^{2}\right)\right|=\frac{1}{2}\left(-10 t^{2}+10 t+60\right) \end{aligned} $

$=-5\left(\mathrm{t}^{2}-\mathrm{t}-6\right)=-5 \mathrm{t}-\frac{1}{2}^{2}+30+\frac{5}{4} \Rightarrow A=-5(t-1 / 2)^{2}+\frac{125}{4}$

Area is maximum when $\mathrm{t}=\frac{1}{2}$

Coordinates of $\mathrm{R} \frac{1}{4}, 1$

Answer: c

3. Minimum area of circle which touches the parabola’s $y=x^{2}+1$ and $y=x^{2}-1$ is

(a) $ \frac{9 \pi}{32}$ sq.unit

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

(c) $ \frac{7 \pi}{32}$ sq.unit

(d) $\frac{9 \pi}{16}$ sq.unit

Show Answer

Solution: $\mathrm{y}=\mathrm{x}^{2}+1$ and $\mathrm{y}^{2}=\mathrm{x}+1$ are symmetrical about $\mathrm{y}=\mathrm{x}$

tangent at point. $A$ and $B$ are parallel to the line $y=x$

$\mathrm{y}=\mathrm{x}^{2}+1$

$\frac{\mathrm{dy}}{\mathrm{dx}}=2 \mathrm{x}=1$

$\mathrm{x}=\frac{1}{2}$

$\mathrm{y}=\frac{5}{4}$ A $\frac{1}{2}, \frac{5}{4}$ B $\frac{5}{4}, \frac{1}{2}$

$\mathrm{AB}=\sqrt{\frac{1}{2}-\frac{5}{4}^{2}+\frac{5}{4}-\frac{1}{2}^{2}}=\frac{3 \sqrt{2}}{4}$

$ y^{2}=x-1 $

$\mathrm{y}=\frac{1}{2}$

$\mathrm{x}=\frac{5}{4} \quad \frac{\mathrm{m}^{2}}{4}$

Area of circle $=\pi \mathrm{r}^{2}=\pi{\frac{3 \sqrt{2}^{2}}{8}}^{2}=\frac{9 \pi}{32}$ sq.unit

Answer: a

4. The equation of the common tangents to the parabola $y=x^{2}$ and $y=-(x-2)^{2}$ is / are

(a) $y=4(x-1)$

(b) $y=0$

(c) $y=-4(x-1)$

(d) $y=-30 x-50$

5. $(3,0)$ is the point from which three normals are drawn to the parabola $y^{2}=4 x$ which meet the parabola in the points $\mathrm{P}, \mathrm{Q}$ and $\mathrm{R}$ then

Comprehension based Questions (Exampels 6 to 8)

Comprehension: 1

Consider the circle $\mathrm{x}^{2}+\mathrm{y}^{2}=9$ and the parabola $\mathrm{y}^{2}=8 \mathrm{x}$. They intersect at $\mathrm{P}$ and $\mathrm{Q}$ in the first and the fourth quadrants, respectively. Tangents to the circle at $P$ and $Q$ intersect the $x$-axis at $R$ and tangents to the parabola at $\mathrm{P}$ and $\mathrm{Q}$ intersect the $\mathrm{x}$-axis at $\mathrm{S}$.

6. The ratio of the area of the triangles $\mathrm{PQS}$ and $\mathrm{PQR}$ is

(a) $1: \sqrt{2}$

(b) $1: 2$

(c) $1: 4$

(d) $1: 8$

Show Answer

Solution: Point of intersection of circle & parabola

$\mathrm{x}^{2}+8 \mathrm{x}-9=0$

$(\mathrm{x}+9)(\mathrm{x}-1)=0$

$\mathrm{X}=1,-9$ (not possible)

$\mathrm{Y}= \pm \mathbf{2} \sqrt{2}$

$\mathrm{P}(1,2 \sqrt{2})$,

Q $(1,-2 \sqrt{2})$

Tangent to the parabola at $P$ is $2 \sqrt{2 y}=4(x+1)$

$\mathrm{S}(-1,0)$

Tangent to the circle at $P$ is $\quad x+2 \sqrt{2 y}=9$

$ \mathrm{R}(9, 0) $

$\frac{\operatorname{ar} \triangle \mathrm{PQS}}{\operatorname{ar} \triangle \mathrm{PQR}}=\frac{\frac{1}{2} \mathrm{xPQxST}}{\frac{1}{2} \mathrm{xPQxRT}}=\frac{\mathrm{ST}}{\mathrm{RT}}=\frac{2}{8}=\frac{1}{4}$

Answer: c

7. The radius of the circum circle of the triangle PRS is

(a) $5$

(b) $3 \sqrt{3}$

(c) $3 \sqrt{2}$

(d) $2 \sqrt{3}$

8. The radius of the in circle of the triangle $\mathrm{PQR} $ is

(a) 4

(b) 3

(c) $\frac{8}{3}$

(d) 2

Comprehension based Questions (Exampels 9 to 11)

Comprehension: 2

If $y=x$ is tangent to the parabola $y=a x^{2}+c$

9. If $a=2$, then the value of $c$ is

(a) $\frac{1}{2}$

(b) $\frac{1}{4}$

(c) $\frac{1}{8}$

(d) $1$

10. If $(1,1)$ is point of contact then ’ $a$ ’ is

(a) $1$

(b) $\frac{1}{2}$

(c) $\frac{1}{3}$

(d) $\frac{1}{4}$

11. If $c=2$ then point of contact is

(a) $(4,4)$

(b) $(2,2)$

(c) $(8,8)$

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

Practice questions

1. The point $\mathrm{P}$ on the parabola $\mathrm{y}^{2}=4 \mathrm{ax}$ for which $|\mathrm{PR}-\mathrm{PQ}|$ is maximum, where $\mathrm{R}(-\mathrm{a}, 0), \mathrm{Q}(0, \mathrm{a})$ is

(a) $(4 a,-4 a)$

(b) $(4 a, 4 a)$

(c) $(a, 2 a)$

(d) $(\mathrm{a},-2 \mathrm{a})$

2. The shortest distance between the parabola $y^{2}=4 x$ and the circle $x^{2}+y^{2}+6 x-12 y+20=0$ is

(a) $4 \sqrt{2}-5$

(b) $4 \sqrt{2}+5$

(c) $3 \sqrt{2}+5$

(d) $3 \sqrt{2}-5$

3. If normals are drawn from a point $p(h, k)$ to the parabola $y^{2}=4 a x$, then the sum of the intercepts which the normals act off from the axis of the parabola is

(a) $4(\mathrm{~h}+0)$

(b) $3(\mathrm{~h}+\mathrm{c})$

(c) $2(\mathrm{~h}+\mathrm{a})$

(d) $(h+a)$

4. If $a \neq 0$ and the line $2 p x+3 q y+4 r=0$ passes through the points of intersection of the parabolas $y^{2}=4 a x$ and $x^{2}=4 a y$, then

(a) $\mathrm{r}^{2}+(3 \mathrm{p}+2 \mathrm{q})^{2}=0$

(b) $ r^{2}+(2 p+3 q)^{2}=0$

(c) $ r^{2}+(3 p-2 q)^{2}=0$

(d) $ r^{2}+(2 p-2 q)^{2}=0$

5. The equation of the tangent at the vertex of the parabola $x^{2}+4 x+2 y=0$ is

(a) $x=2$

(b) $\mathrm{x}=-2$

(c) $y=-2$

(d) $y=2$

6. The common tangent to the parabolas $y^{2}=4 a x$ and $x^{2}=4 a x$ and $x^{2}=32 a y$ is

(a) $x+2 y-4 a=0$

(b) $x+2 y+4 a=0$

(c) $x-2 y+4 a=0$

(d) $x-2 y-4 a=0$

7. The shortest distnae between the parabolas $y^{2}=4 x$ and $y^{2}=2 x-6$ is

(a) $\sqrt{5}$

(b) $2$

(c) $3$

(d) none of these

8. The largest value of a for which the circle $x^{2}+y^{2}=a^{2}$ falls totally in the interior of the parabola $y^{2}=4(x+4)$ is

(a) 4

(b) $4 \sqrt{3}$

(c) $3 \sqrt{3}$

(d) $2 \sqrt{3}$

Multiple choice questions with one or more than one correct answer.

9. Let $P\left(x _{1}, y _{1}\right)$ and $Q\left(x _{2}, y _{2}\right), y _{1}<0, y _{2}, 0$, be the end points of the latus rectum of the ellipse $x^{2}+4 y=4$. The equations of parabolas with latus rectum $\mathrm{PQ}$ are

(a) $x^{2}+2 \sqrt{3} y=3+\sqrt{3}$

(b) $x^{2}-2 \sqrt{3} y=3+\sqrt{3}$

(c) $x^{2}+2 \sqrt{3} y=3-\sqrt{3}$

(d) $x^{2}-2 \sqrt{3} y=3-\sqrt{3}$

10. The tangent $\mathrm{PT}$ and the normal $\mathrm{PN}$ to the parabola $\mathrm{y}^{2}=4 \mathrm{ax}$ at a point $\mathrm{P}$ on it meet its axis at point $\mathrm{T}$ 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 $(a, 0)$

Match the following :

11. Consider the parabola $\mathrm{y}^{2}=12 \mathrm{x}$

Assertion and Reasoning

12. Statement 1 : The curve $y=-\frac{x^{2}}{2}+x+1$ is symmetric with respect to the line $x=1$.

Statement 2 : A parabola is symmetric about its axis.

(A) Statement 1 is True, Statement 2 is True; Statement 2 is a correct explanations for statement 1.

(B) Statement 1 is True, statement 2 is true, statement 2 is not a correct explanation for statement 1 .

(C) Statement 1 is true, statement 2 is false.

(D) Statement 1 is false, statement 2 is true.