Intersection of a Line and Finding Roots of a Parabola

Q.

The locus of the mid-point of the line segment joining the focus to a moving point on the parabola $y^2=4ax$ is another parabola with directrix

1. $x=-a$

2. $x=\frac{-a}{2}$

3. $x=0$

4. $x=\frac{a}{2}$

Q. If the line $ax+y=c,$ touches both the curves $x^2+y^2=1$ and $y^2=4\sqrt{2}x,$ then $|c|$ is equal to :

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

Q.

The straight line $y=2x+\lambda$ does not meet the parabola $y^2=2x$, if

1. $\lambda =1$

2. $\lambda =4$

3. $\lambda <\frac{1}{4}$

4. $\lambda >\frac{1}{4}$

Q.

Find the point of intersection of line x/a+y/b=1, x/b+y/a=1(a≠±b)

Q. The equation of the tangent to the parabola y = $x^2$ - 2x + 3 at the point (2, 3) is .

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

Q. If the line y = x + 2a touches the parabola $y^2=4a(x+a)$, then the point of contact is .

1. (0, a)
2. (0, 2a)
3. (a, 0)
4. (2a, 0)

Q.

The line 2x-y+4=0 cuts the parabola $y^2=8x$ in P and Q.The mid-point of PQ is

1. (-1, -2)

2. (1, 2)

3. (1, -2)

4. (-1, 2)

Q.

Write the formula to find the vertex of the parabola.

Q. $A$ is a point on either of two lines $y+\sqrt{3}|x|=2$ at a distance of $\frac{4}{\sqrt{3}}$ units from their point of intersection. The coordinates of the foot of perpendicular from $A$ on the bisector of the angle between them are

Q.

Find the equation of a line perpendicular to the line $\sqrt{3}x-y+5=0$ and at a distance of 3 units from the origin.

Q. A variable line $y=l\left(x\right)$ intersects the parabola $y=x^2$ at points $P$ and $Q$ whose $x$-coordinates are $\alpha$ and $\beta$ respectively with $\alpha <\beta$. The area of the figure enclosed by the segment $PQ$ and the parabola is always equal to $\frac{4}{3}$. The variable segment $PQ$ has its middle point as $M$. Then the value of $(\beta -\alpha )$ is greater than (or) equal to-

1. $1$
2. $2$
3. $4$
4. $8$

Q. Let S be the focus of the parabola $y^2=8x$ and $PQ$ be the common chord of the circle $x^2+y^2-2x-4y=0$ and the given parabola. The area of the $\Delta OPS$ ($O$ is the origin) is ___.

Q.

Let the curve $C$ be the mirror image of the parabola $y^2=4x$ with respect to the line $x+y+4=0$. If $A$ and $B$ are the points of intersection of $C$ with the line $y=-5$, then the distance between $A$ and $B$ is___

Q. Find the equation of the line which is intersecting the $y$-axis at a distance of $2$ units above the origin and making an angle of ${30}^{\circ }$ with positive direction of the $x$-axis.

Q. Write the following numerals using Indian System words or international system (as required in words):
5, 420, 700 = ___

Q. Find all the points on x+y=2, which are at a distance of $\frac{4}{5}$ units from 4x+3y=7.

1. (5, -3)

2. (-3, 5)

3. (6, -4)

4. (-4, 6)

Q. The sum of the intercepts on the axes of the tangent to the curve $\sqrt{x}+\sqrt{y}=3$ at $(4,1)$ is

1. $7$
2. $9$
3. $3$
4. $5$

Q. Let $P(a{t}^2,2at),Q(a{r}^2,2ar)$ be three points on a parabola $y^2=4ax$. If $PQ$ is the focal chord and $PK,QR$ are parallel where the co-ordinates of $K$ is $(2a,0)$, then the value of $r$ is

1. $\frac{t}{1-t^2}$
2. $\frac{1-t^2}{t}$
3. $\frac{t^2+1}{t}$
4. $\frac{t^2-1}{t}$

Q.

The line y = mx + 1 is a tangent to the curve y² = 4x if the value of m is

(A) 1 (B) 2 (C) 3 (D)

Q. The sum of the coordinates of the point of contact of the line $x-2y=1$ with parabola $y^2=2(x-3)$ is

Q.

If the tangent at the point P(2, 4) to the parabola $y^2=8x$ meets the parabola $y^2=8x+5$ at Q and R,
then the midpoint of QR is

1. (2, 4)

2. (7, 9)

3. (3, 3)

4. (4, 2)

Q. The normals to the parabola $y^2=4x$ at the points $P,Q$ and $R$ are concurrent at the point $(15,12)$. The coordinates of the centroid of the triangle $PQR$ are

1. $(0,\frac{26}{3})$
2. $\left(\frac{8}{3},\frac{2}{3}\right)$
3. $\left(\frac{26}{3},0\right)$
4. $\left(\frac{13}{3},\frac{1}{2}\right)$

Q. A line coming from point $\left(3,4\right)$ is reflected on a point A of x-axis and it passes through point $\left(1,8\right)$ then coordinate of point A is

1. $\left(7/3,0\right)$
2. $\left(5,0\right)$
3. $\left(29/3,0\right)$
4. $\left(9/2,0\right)$

Q. Find the equation of the plane through the point $2\hat{i}+\hat{j}-\hat{k}$ and passing through the line of intersection of the planes $\overrightarrow{r}·\left(\hat{i}+3\hat{j}-\hat{k}\right)=0\mathrm{and}\overrightarrow{r}·\left(\hat{j}+2\hat{k}\right)=0.$

Q. Let $B(16,-16)$ be an end point of a focal chord $AB$ of the parabola $y^2=16x$ and let the tangent and normal at $A$ meet the axes in $T$ and $N$ respectively. Then reflection of the circle through $A,T$ and $N$ about the line $y=x$, is

1. $x^2+y^2-8x+9=0$
2. $x^2+y^2-8y-9=0$
3. $x^2+y^2+8x+9=0$
4. $x^2+y^2+8x-9=0$

Q. Show that the line x + y = 4 intersects the parabola $y^2=2x$ at two points P and Q. Find the coordinates of these points and the length PQ.

Q. Let $S$ be the focus of the parabola $y^2=8xandletPQ$ be the common chord of the circle $x^2+y^2-2x-4y=0$ and the given parabola. The area of the triangle $PQS$ is

Q. If the line y = x + 2a touches the parabola $y^2=4a(x+a)$, then the point of contact is .

1. (0, a)
2. (0, 2a)
3. (a, 0)
4. (2a, 0)

Q. The length of the straight line x − 3y = 1 intercepted by the hyperbola x² − 4y² = 1 is

(a)

(b)

(c)

(d) none of these

Q. Find the area of the triangle formed by the points $(0,0),(1,0),(0,1)$.