Intersection of a Line and Finding Roots of a Parabola

Q.

Which pair of points lie on the same side of $3x-8y-7=0$

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

2. $\left(4,-3\right)and\left(0,1\right)$

3. $\left(-3,-4\right)and\left(1,2\right)$

4. $\left(-1,-1\right)and\left(3,7\right)$

Q.

How do you find the equation of a parabola with $3$ points $?$

Q.

For which of the following curves, the line $x+\sqrt{3}y=2\sqrt{3}$ is the tangent at the point $(\frac{3\sqrt{3}}{2},\frac{1}{2})$?

1. $x^2+9y^2=9$

2. $2x^2-18y^2=9$

3. $y^2=\frac{x}{6\sqrt{3}}$

4. $x^2+y^2=7$

Q.

Dilate the figure by a scale factor of $0.5$ with the origin as the center of dilation. What are the vertices of the image?

1. $\text{F}\left(3,2.5\right),\text{G}\left(5,0.5\right),\text{H}\left(2,1\right)$

2. $\text{F}\left(2.5,3\right),\text{G}\left(0.5,5\right),\text{H}\left(0.5,1\right)$

3. $\text{F}\left(3,2.5\right),\text{G}\left(5,0.5\right),\text{H}\left(1,0.5\right)$

4. $\text{F}\left(-3,-2.5\right),\text{G}\left(-5,-0.5\right),\text{H}\left(-1,-0.5\right)$

Q.

The equation of the tangent to the curve$y=x+\frac{4}{x^2}$, that is parallel to the $x$-axis is

1. $y=0$

2. $y=1$

3. $y=2$

4. $y=3$

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. Let $P$ and $Q$ be distinct points on the parabola $y^2=2x$ such that a circle with $PQ$ as diameter passes through the vertex $O$ of the parabola. If $P$ lies in the first quadrant and the area of the triangle $△OPQ$ is $3\sqrt{2}\text{sq. units}$, then which of the following is/are the coordinates of $P$?

1. $(4,2\sqrt{2})$
2. $(9,3\sqrt{2})$
3. $(\frac{1}{4},\frac{1}{\sqrt{2}})$
4. $(1,\sqrt{2})$

Q.

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

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

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

3. $\lambda =4$

4. $\lambda =1$

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. (4, 2)

3. (7, 9)

4. (3, 3)

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