Equation of Pair of Tangents: Hyperbola

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.

The equations of the tangents drawn to the curve y² - 2x³ - 4y + 8 = 0 from the point (1, 2) is... .

1. 

2. 

3. 

4. 

Q.

Show that the line x+y=1 touches the parabola y=x-x²

Q.

The equation of tangent to the curve y = x³ + 2x + 6 which is perpendicular to the line x + 14y + 4 = 0 is :

1. 14x- y + 22 = 0

2. 14x- y + 18 = 0

3. 14x - y + 5 = 0

4. 14x -y + 1= 0

Q. Tangents are drawn from any point on the circle $x^2+y^2=41$ to the Ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ then the angle between the two tangents is

1. 
2. 
3. 
4. 

Q. Find the equation of the tangent to x²/a² -y²/b² = 1 at (x₁, y₁)

Q.

Find the points on the curve y = x³ at which the slope of the tangent is equal to the y-coordinate of the point.

Q. The equation of the pair of tangents drawn from the point (4, 3) to the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ is .

1. $3x^2-4\mathrm{xy}-12x+16y=0$
2. $3x^2-2\mathrm{xy}-12x+16y=0$
3. $3x^2-8\mathrm{xy}-12x+16y=0$
4. $3x^2+4\mathrm{xy}-12x+16y=0$

Q.

Find the equation of the tangents drawn form the point (5, 3) to the hyperbola $\frac{x^2}{25}-\frac{y^2}{9}=1.$

1. $3x^2-5xy-15x+25y=0$

2. $3x^2-5xy+10x+15y=0$

3. $2x^2-5xy-10x+15y=0$

4. $2x^2-5xy-15x+25y=0$

Q. If tangents are drawn from any point on the circle $x^2+y^2=25$ to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1,$ then the angle between the tangents is

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

Q. Tangents are drawn to the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1,$ parallel to the straight line $2x–y=1.$ The point of contact of the tangents on the hyperbola are

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

Q. Equation of a common tangent to the curves and
$y^2=8xandxy=-1$ is

1. y = x +2
2. 3y = 9x +2
3. y = 2x +1
4. 2y = x + 8

Q. The equation of the pair of tangents drawn from the point (4, 3) to the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ is .

1. $3x^2-4\mathrm{xy}-12x+16y=0$
2. $3x^2-2\mathrm{xy}-12x+16y=0$
3. $3x^2-8\mathrm{xy}-12x+16y=0$
4. $3x^2+4\mathrm{xy}-12x+16y=0$

Q.

Find the equation of tangent to the curve y = (x-1)² which is parallel to the chord joining (1, 0) and (3, 4)

1. 2x + y + 3 = 0

2. 2x - y - 3 = 0

3. x - y = 1

4. x + y = 3

Q. Find the equation of the tangent line to the curve y = x² + 4x − 16 which is parallel to the line 3x − y + 1 = 0.

Q. Write the coordinates of the point on the curve y² = x where the tangent line makes an angle $\frac{\mathrm{\pi }}{4}$ with x-axis.

Q. The line $4x-7y+10=0$ intersects the parabola , $y^2=4ax$ at the points A & B . The co-ordinates of the point of intersection of the tangents drawn at the points A & B are :

1. $\left(\frac{7}{2},\frac{5}{2}\right)$
2. $\left(\frac{5}{3},\frac{8}{3}\right)$
3. $\left(\frac{5}{2},\frac{7}{2}\right)$
4. $\left(-\frac{7}{2},-\frac{5}{2}\right)$

Q. If $I={\int }_2^3\frac{2x^5+x^4-2x^3+2x^2+1}{(x^2+1)(x^4-1)}dx$ then $I$equals

1. $\frac{1}{2}\log 6+\frac{1}{10}$
2. $\frac{1}{2}\log 6-\frac{1}{10}$
3. $\frac{1}{2}\log 2+\frac{1}{10}$
4. $\frac{1}{2}\log 3-\frac{1}{10}$

Q. find the equation of the tangents to the parabola y²= 6xwhich pass through the point (3/2, 5). and also find the equation of the tangents to the parabola y²+12x=0 from the point (3, 8).

Q. Let $h\left(x\right)=min(x,x^2)$, for every real numbers $x$. Then

1. $h$ is not continuous for all $x$
2. $h$ is differentiable for all $x$
3. $h^{\prime }\left(x\right)\ne 1$, for all $x>1$
4. $h$ is not differentialble at two values of $x$

Q. Tangents to the parabola at the extremities of a common chord $AB$ of the circle $x^2+y^2=5$ and the parabola $y^2=4x$ intersect at the point $T$. A square $ABCD$ is constructed on this chord lying inside the parabola, then $\frac{{[{\left(TC\right)}^2+{\left(TD\right)}^2]}^2}{1600}$ is equal to

Q. Tangents are drawn from any point on the circle $x^2+y^2=41$ to the Ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ then the angle between the two tangents is

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

Q. Determine the equation(s) of tangent (s) line to the curve y = 4x³ − 3x + 5 which are perpendicular to the line 9y + x + 3 = 0.

Q. Find the coordinates of the point on the curve y² = 3 − 4x where tangent is parallel to the line 2x + y − 2 = 0.

Q.

Find the equation of the tangents drawn form the point (5, 3) to the hyperbola $\frac{x^2}{25}-\frac{y^2}{9}=1.$

1. $3x^2-5xy-15x+25y=0$

2. $3x^2-5xy+10x+15y=0$

3. $2x^2-5xy-10x+15y=0$

4. $2x^2-5xy-15x+25y=0$

Q. TANGENT AND NORMAL
14. If the tangent to the curve y = $x^3$ + ax + b at (1, —6) is parallel to the line x — y + 5 = O, find a and b.

Q.

Find the equation of the tangents drawn form the point (5, 3) to the hyperbola $\frac{x^2}{25}-\frac{y^2}{9}=1.$

1. $3x^2-5xy-15x+25y=0$

2. $3x^2-5xy+10x+15y=0$

3. $2x^2-5xy-10x+15y=0$

4. $2x^2-5xy-15x+25y=0$

Q. Let the tangent to the graph of $y=f\left(x\right)$ at the point $x=a$ be parallel to the x-axis, and let $f^{\prime }(a-h)>0$ and $f^{\prime }(a+h)<0$, where h is a very small positive number.Then the ordinate of the point is

1. a maximum
2. a minimum
3. both a maximum and minimum
4. neither a maximum nor a minimum

Q. x^2/3+y^2/3=2 at(1, 1) find the equation of the tangent and normal line of the following curve

Q. At what points will be tangents to the curve y = 2x³ − 15x² + 36x − 21 be parallel to x-axis? Also, find the equations of the tangents to the curve at these points.