Tangent To a Parabola

Q.

If the area of the triangle formed by the positive x-axis, the normal and the tangent to the circle ${(x-2)}^2+{(y-3)}^2=25$ at the point $(5,7)$ is $A$, then $24A$ is equal to

Q.

The tangents from origin to the parabola $y^2+4=4x$ are inclined at.

1. 0

2. 

3. 

4. 

Q. Slope of tangent to $x^2=4y$ from (-1, -1) can be

1. $\frac{-1\pm \sqrt{5}}{2}$
2. $\frac{-3-\sqrt{5}}{2}$
3. $\frac{1-\sqrt{5}}{2}$
4. $\frac{1+\sqrt{5}}{2}$

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. The slope of aline is double of the slope of another line. If tangent of the angle between hem is $\frac{1}{3}$. Find the slopes of the lines.

1. -1 and -2

2. and -1

3. 1 and 2

4. and 1

Q.

In previous question, we have shown that the locus of the feet of the perpendicular draw from the focus of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ upon ay tangent is its auxiliary circle $x^2+y^2=a^2$ then the product of these perpendiculars from the focusupon ay tangent is _____

1. 

2. 

3. 

4. 

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. Let $P$ be the point of intersection of the common tangents to the parabola $y^2=12x$ and the hyperbola $8x^2-y^2=8.$ If $S$ and $S^{\prime }$ denote the foci of the hyperbola where $S$ lies on the positive $x-$axis then $P$ divides $S{S}^{\prime }$ in a ratio

1. $14:13$
2. $13:11$
3. $5:4$
4. $2:1$

Q. Two tangents are drawn from the point $(-2,-1)$ to the parabola $y^2=4x$. If $\theta$ is the angle between these tangents then $\tan \theta$ equals to

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

Q.

Tangents drawn at A$(a{t}_1^2,2a{t}_1)$ and B $(a{t}_2^2,2a{t}_2)$ intersect at (p, q).

Then p and q are ____and ____ of corresponding x and y coordinates of A and B respectively $(given{t}_1\ne t_2)$

1. AM, GM

2. AM, HM

3. GM, HM

4. GM, AM

Q. Consider the parabola whose focus at $(0,0)$ and tangent at vertex is $x-y+1=0$.
Tangents drawn to the parabola at the extremities of the chord $3x+2y=0$ intersects at an angle

1. $\frac{\pi }{6}$
2. $\frac{\pi }{3}$
3. $\frac{\pi }{2}$
4. $\frac{\pi }{4}$

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. 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. $y^2=4x$ and $y^2=-8(x-a)$ intersect at point $A$ and $C$. Points $O(0,0),A,B(a,0),C$ are concyclic.

The length of common chord of parabolas is

1. $2\sqrt{6}$
2. $4\sqrt{3}$
3. $6\sqrt{5}$
4. $8\sqrt{2}$

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 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(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. A tangent is drawn to parabola $y^2-4x+4=0$ at a point $P$ which cuts the directrix at the point $Q$. If a point $R$ is such that it divides $QP$ externally in ratio $1:2$, then the locus of point $R$ is

1. $y(x+1{)}^2+4=0$
2. $y^2(x+1)+4=0$
3. $y=0$
4. $(x+1)(y-1{)}^2-4=0$

Q. Consider the parabola whose focus at $(0,0)$ and tangent at vertex is $x-y+1=0$.
The equation of the parabola is

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

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. $2$