Tangent To a Parabola

Q.

A line parallel to the straight line $2x-y=0$ is tangent to the hyperbola $\frac{x^2}{4}-\frac{y^2}{2}=1$ at the point$(x_1,y_1)$.

Then ${x_1}^2+5{y_1}^2$ is equal to

1. $6$

2. $10$

3. $8$

4. $5$

Q. If the tangent to the parabola $y^2=x$ at a point $(\alpha ,\beta ),(\beta >0)$ is also a tangent to the ellipse, $x^2+2y^2=1$, then $\alpha$ is equal to :

1. $2\sqrt{2}-1$
2. $2\sqrt{2}+1$
3. $\sqrt{2}+1$
4. $\sqrt{2}-1$

Q. The equation of common tangent to the curves $y^2=16x$ and $xy=–4$, is :

1. $x-2y+16=0$
2. $x+y+4=0$
3. $2x-y+2=0$
4. $x-y+4=0$

Q. A line is a common tangent to the circle $(x–3{)}^2+y^2=9$ and the parabola $y^2=4x$. If the two points of contact $(a,b)$ and $(c,d)$ are distinct and lie in the first quadrant, then $2(a+c)$ is equal to

Q. Equation of a common tangent to the parabola $y^2=4x$ and the hyperbola $xy=2$ is:

1. $x-2y+4=0$
2. $x+y+1=0$
3. $4x+2y+1=0$
4. $x+2y+4=0$

Q. The slope of the line touching both the parabolas $y^2=4x$ and $x^2=-32y$ is

1. $\frac{1}{2}$
2. $\frac{3}{2}$
3. $\frac{1}{8}$
4. $\frac{2}{3}$

Q. A parabola $y=a{x}^2+bx+c$ crosses the x-axis at $(\alpha ,0)(\beta ,0)$ both to the right of the origin. A circle also passes through these two points. The length of a tangent from the origin to the circle is

1. $\sqrt{\frac{bc}{a}}$
2. $\sqrt{\frac{c}{a}}$
3. $a{c}^2$
4. $\frac{b}{a}$

Q. The equation of the latus rectum of a parabola is $x+y=8$ and the equation of the tangent at the vertex is $x+y=12$. Then the length of the latus rectum is

1. $8\text{units}$
2. $8\sqrt{2}\text{units}$
3. $2\sqrt{2}\text{units}$
4. $4\sqrt{2}\text{units}$

Q. If the tangents to the parabola $x=y^2+c$ from origin are perpendicular, then $c$ is equal to

1. $\frac{1}{2}$
2. $1$
3. $\frac{1}{4}$
4. $2$

Q. Tangents are drawn from the point $(-8,0)$ to the parabola $y^2=8x$ touch the parabola at $P$ and $Q$. If $F$ is the focus of the parabola, then the area of the triangle $PFQ$ (in sq. units) is equal to

1. $64$
2. $24$
3. $32$
4. $48$

Q. Consider the parabola $y^2=8x$. Let ${\Delta }_1$ be the area of the triangle formed by the endpoints of its latus rectum and the points $P(\frac{1}{2},2)$ on the parabola and ${\Delta }_2$ be the area of the triangle formed by drawing tangents at P and at the endpoints of the latus rectum. Then, $\frac{{\Delta }_1}{{\Delta }_2}$ is ___

Q. Two parabolas with a common vertex and with axes along $x-$ axis and $y-$ axis, respectively, intersect each other in the first quadrant. If the length of the latus rectum of each parabola is $3$, then the equation of the common tangent to the two parabolas is :

1. $4(x+y)+3=0$
2. $3(x+y)+4=0$
3. $x+2y+3=0$
4. $8(2x+y)+3=0$

Q.

The portion of a tangent to a parabola cut off between the directrix and the curve subtends right angle at the focus.

1. True

2. False

Q. The straight line $x+y=k+1$ touches the parabola $y=x(1-x)$ if

1. $k=-1$
2. $k=0$
3. $k=1$
4. $k$ takes any real value

Q.

The equation of four circles are $(x\pm a{)}^2+(y\pm a{)}^2=a^2$ . The radius of a circle touching all the four circles is

1. 

2. 

3. 

4. None of these

Q. The equation(s) of tangents to the parabola $y^2$ = 12x, which passes through the point (2, 5) is/are:

1. y = x + 3
2. y = x - 3
3. 2y = 3x - 4
4. 2y = 3x + 4

Q. Consider two concentric circles $C_1:x^2+y^2-4=0$ and $C_2:x^2+y^2-9=0.$ A parabola is drawn through the points where $C_1$ meets $y-$axis and having an arbitrary tangent of $C_2$ as its directrix. If $C$ is the curve of locus of focus of drawn parabola and $e$ is the eccentricity of the curve $C,$ then the value of $12e$ is

Q. The tangent to the parabola $y=x^2+ax+1$ at the point of intersection of $y-$axis also touches the circle $x^2+y^2=r^2$ and no point of the parabola is below $x-$axis. Then

1. the radius of circle when ${}^{\prime }{a}^{\prime }$ attains its maximum value is $\frac{1}{\sqrt{10}}$
2. the slope of the tangent when radius of the circle is maximum is $0$
3. area bounded by the tangent and the coordinate axes is minimum when $a=1$
4. the minimum area bounded by the tangent and the coordinate axes is $\frac{1}{4}$ sq. unit

Q. Consider the parabola $(x-1{)}^2+(y-2{)}^2=\frac{(12x-5y+3{)}^2}{169}$
$\begin{array}{cc}ColumnI& ColumnII\\ \text{a. Locus of point of intersection of perpendicular tangent}& \text{p.}(12x-5y-2=0)\\ \text{b. Locus of foot of perpendicular from focus upon any tangent}& \text{q.}(5x+12y-29=0)\\ \text{c. Line along which minimum length of focal chord occurs}& \text{r.}(12x-5y+3=0)\\ \text{d. Line about which parabola is symmetrical}& \text{s.}(24x-10y+1=0)\end{array}$

1. $\text{a}-\text{r},\text{b}-\text{s},\text{c}-\text{p},\text{d}-\text{q}$
2. $\text{a}-\text{s},\text{b}-\text{r},\text{c}-\text{p},\text{d}-\text{q}$
3. $\text{a}-\text{r},\text{b}-\text{q},\text{c}-\text{p},\text{d}-\text{s}$
4. $\text{a}-\text{p},\text{b}-\text{s},\text{c}-\text{q},\text{d}-\text{r}$

Q. The line $x+y=1$ touches the parabola $y^2-y+x=0$ at the point

1. $(1,1)$
2. $(\frac{1}{2},\frac{1}{2})$
3. $(0,1)$
4. $(1,0)$

Q. Two straight lines are perpendicular to each other. One of them touches the parabola $y^2=4a(x+a)$ and the other touches $y^2=4b(x+b)$. Their point of intersection lies on the line

1. $x-a+b=0$
2. $x+a-b=0$
3. $x+a+b=0$
4. $x-a-b=0$

Q.

Let $P$ and $Q$ are the points on the parabola $y^2=4x,$ if so that the line segment $PQ$ subtends a right angle at the vertex. If $PQ$ intersects the axis of the parabola at $R$, then the distance of the vertex from $R$ is

1. $1$

2. $2$

3. $4$

4. $6$

Q. The area of the region bounded by the curve $y=x^3$, its tangent at $(1,1)$ and $x$-axis is

1. $\frac{1}{12}$
2. $\frac{2}{17}$
3. $\frac{1}{6}$
4. $\frac{2}{15}$

Q. If tangents are drawn at the points $A(1,2),B(4,-4)$ and a variable point ${}^{\prime }{C}^{\prime }$ lies on the parabola $y^2=4x$, then the locus of the orthocentre of triangle formed by these tangents is

1. $x=0$
2. $x-1=0$
3. $x+1=0$
4. $y=0$

Q.

The curve given by $x+y=e^{xy}$has a tangent parallel to the$y$- axis at the point :

1. $\left(0,1\right)$

2. $\left(1,1\right)$

3. $\left(0,0\right)$

4. $\left(1,0\right)$

Q. The common tangents to the circle $x^2+y^2=2$ and the parabola $y^2=8x$ touch the circle at the points $P,Q$ and the parabola at the points $R,S$. Then the area of the quadrilateral $PQRS$ is

1. $3$
2. $6$
3. $9$
4. $15$

Q. If $\alpha$ is the inclination of a tangent to the parabola $y^2=4ax$ then the distance between the tangent and a parallel normal is

1. 
2. 
3. 
4. 

Q. The equation(s) of tangents to the parabola $y^2$ = 12x, which passes through the point (2, 5) is/are:

1. y = x + 3
2. y = x - 3
3. 2y = 3x - 4
4. 2y = 3x + 4

Q. If a chord of solpe $m$ of the circle $x^2+y^2=4$ touches the parabola $y^2=4x$, then
(where [.] represents G.I.F)

1. least value of $\left[m^2\right]=0$
2. least value of $\left[m\right]=0$
3. $m\in (-\infty ,-\sqrt{\frac{\sqrt{2}-1}{2}})\cup (\sqrt{\frac{\sqrt{2}-1}{2}},\infty )$
4. $m\in (-\sqrt{\frac{\sqrt{2}-1}{2}},\sqrt{\frac{\sqrt{2}-1}{2}})$

Q. The equation of common tangent to the curves $y^2=16x$ and $xy=–4$, is :

1. $x-2y+16=0$
2. $x+y+4=0$
3. $2x-y+2=0$
4. $x-y+4=0$