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

(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 $y^2=4a(x-b)16=4a(4-b)$

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

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

Let $R$ be $(t^2,2t)$ on the Parabola thus

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

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

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 $\frac{5}{4},\frac{1}{2}$

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

$y^2=x-1$

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

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

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) $y=-30x-50$

5. $(3,0)$ is the point from which three normals are drawn to the parabola $y^2=4x$ 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 $2\sqrt{2y}=4(x+1)$

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

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

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

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

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) $(4a,-4a)$

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

(c) $(a,2a)$

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

2. The shortest distance between the parabola $y^2=4x$ and the circle $x^2+y^2+6x-12y+20=0$ 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 $y^2=4ax$, 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 $a\ne 0$ and the line $2px+3qy+4r=0$ passes through the points of intersection of the parabolas $y^2=4ax$ and $x^2=4ay$, then

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

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

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

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

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

(a) $x=2$

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

(c) $y=-2$

(d) $y=2$

6. The common tangent to the parabolas $y^2=4ax$ and $x^2=4ax$ and $x^2=32ay$ is

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

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

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

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

7. The shortest distnae between the parabolas $y^2=4x$ and $y^2=2x-6$ 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 $P(x_1,y_1)$ and $Q(x_2,y_2),y_1<0,y_2,0$, be the end points of the latus rectum of the ellipse $x^2+4y=4$. 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 $(\frac{2\mathrm{a}}{3},0)$

(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.