Tangent To a Parabola

Q.

Let denote a curve $y=y\left(x\right)$ which is in the first quadrant and let the point $\left(1,0\right)$ lie on it. Let the tangent to at a point $P$ intersect the $y$-axis at $Y_P$. If $P{Y}_P$ has length $1$ for each point $P$ on , then which of the following is options is/are correct?

1. $y={\log }_e\left(\frac{1+\sqrt{1-x^2}}{x}\right)-\sqrt{1-x^2}$

2. $xy-\sqrt{\left[1-x^2\right]}=0$

3. $y={\log }_e\left(\frac{1+\sqrt{1-x^2}}{x}\right)+\sqrt{1-x^2}$

4. $xy+\sqrt{\left[1-x^2\right]}=0$

Q.

The length of sub-tangent to the curve $x^2{y}^2=a^4$ at the point $(-a,a)$ is

1. $3a$

2. $2a$

3. $a$

4. $4a$

Q.

The slope of tangent to the curve $x=t^2+3t–8,y=2t^2–2t–5$at the point $(2,–1)$ is

1. $\frac{22}{7}$

2. $\frac{6}{7}$

3. $-6$

4. None of these

Q.

If the lines joining the origin to the intersection of the line $y=mx+2$ and the circle $x^2+y^2=1$ are at right angles, then:

1. $m=\sqrt{3}$

2. $m=\pm \sqrt{7}$

3. $m=1$

4. $m=\sqrt{5}$

Q.

The area of the triangle formed by the tangent and the normal to the parabola $y^2=4ax$, both drawn at the same end of the latus rectum, and the axis of the parabola is

1. $2\sqrt{2}{a}^2$

2. $2a^2$

3. $4a^2$

4. None of these

Q.

The abscissa of the points of the curve $y=x(x-2)(x-4)$ where tangents are parallel to x-axis is obtained as

1. $x=2\pm \frac{2}{\sqrt{3}}$

2. $x=1\pm \frac{1}{\sqrt{3}}$

3. $x=2\pm \frac{1}{\sqrt{3}}$

4. $x=\pm 1$

Q.

If a tangent to the parabola $y^2=4ax$ meets the x-axis at T and the tangent at the vertex A, at P. Let the rectangle TAPQ be completed. Then the locus of the point Q is

1. a circle

2. a straight line

3. a parabola

4. an ellipse

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

Q.

The point of intersection of the tangents to the parabola $y^2=4x$ at the points where parameter 't' has value 3 and 5.

1. (10, 6)

2. (4, 9)

3. (15, 8)

4. (15, 10)

Q.

Find the tangent of the angle between the lines whose intercepts on the axes are respectively $a$, $-b$ and $b$, $-a$

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 area of cyclic quadrilateral $OABC$ is

1. $24\sqrt{3}$
2. $48\sqrt{2}$
3. $12\sqrt{6}$
4. $18\sqrt{5}$

Q. If tangent at $P$ and at vertex of the parabola $y^2=4ax$ meets at $Q$, then the value of $\angle SQP$, where $S$ is the focus, is

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

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. 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. Two straight lines are perpendicular to each other one of them touches the parabola $y^2=4a(x+a)$ and the other touches $y^2=4b(x+b).$ The locus of the point of intersection of these two lines is

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

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. Two straight lines are perpendicular to each other one of them touches the parabola $y^2=4a(x+a)$ and the other touches $y^2=4b(x+b).$ The locus of the point of intersection of these two lines is

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

Q. The locus of point of intersection of any tangent to the parabola $y^2=4a(x-2)$ with a line perpendicular to it and passing through the focus, is

1. x = 2
2. y = 0
3. x = a
4. x = a + 2

Q.

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

1. 0

2. $\frac{\pi }{3}$

3. $\frac{\pi }{6}$

4. $\frac{\pi }{2}$

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.

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. GM, HM

2. GM, AM

3. AM, GM

4. AM, HM

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. Number of tangents to $y^2=2x$ through $(1,2)$ is

1. $0$
2. $1$
3. $2$
4. $3$

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

1. 0

2. $\frac{\pi }{3}$

3. $\frac{\pi }{6}$

4. $\frac{\pi }{2}$

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.

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. GM, HM

2. GM, AM

3. AM, GM

4. AM, HM

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