Equation of Tangent When a Point Is Given

Q. Let $P$ and $R$ are two points on the parabola $y=x^2$ such that $PS$ and $RS$ be the tangents at $P$ and $R,$ and $PQ$ and $RQ$ be the normal drawn at $P$ and $R$ respectively. If the length of rectangle $PQRS$ form is twice of its width, then which of the following is/are correct?

1. The possible coordinates of $S$ are $(\frac{1}{4},\frac{1}{16})$ or $(-\frac{1}{4},\frac{1}{16})$
2. The possible coordinates of $S$ are $(\frac{3}{8},-\frac{1}{4})$ or $(-\frac{3}{8},-\frac{1}{4})$
3. Area of the rectangle $PQRS$ is $\frac{125}{128}$
4. Length of the rectangle is $\frac{5\sqrt{5}}{8}$

Q. If at x = 1, y = 2x is tangent to the parabola $y=a{x}^2+bx+c$, then respective values of a, b, c possible are

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

Q.

A tangent is drawn at the point $P(\alpha ,\beta )$ on the parabola $y^2=4ax$. This tangent meets a second parabola $y^2=4a(x+k)$ at A and B, what is the midpoint of AB?

1. $(\alpha +k,\beta )$

2. $(\alpha -k,\beta -k)$

3. $[\frac{\alpha }{k},\frac{\beta }{k}]$

4. $(\alpha ,\beta )$

Q. An equation of a tangent drawn to the curve $y=x^2-3x+2$ from the point $(1,-1)$ is

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

Q. A circle touches the parabola $y^2=4x$ at $M(1,2)$ and also touches its directrix. The $y$-coordinate of the point of contact of the circle and the directrix is

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

Q. A tangent is drawn at a point on the parabola $y^2=10x$. If this tangent intersect the parabola $y^2=10x+p$ at 2 points; then,

1. $-10<p<10$
2. $p<0$
3. $p>0$
4. p = 0

Q.

P(t = 2) is a point on the parabola $y^2=4ax$. What is the point of the intersection of tangent at P and the directrix of the parabola?

1. (10, 10)
2. (-10, 15)
3. (15, 10)
4. (0, 0)