Position of a Line W.R.T Hyperbola

Q.

The line $y=mx+c$ becomes a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, then the value of c is

1. $\pm \sqrt{a^2{m}^2+b^2}$

2. $\pm \sqrt{a^2{m}^2-b^2}$

3. $\pm \sqrt{a^2{m}^2+a^2}$

4. $\pm \sqrt{a^2{m}^2-a^2}$

Q. LOLOq-pthe line 3x-4y + 5 =0 is a tangent to the parabola y2 = 4ax, then a is equal to

Q.

The tangent of the angle which the part of the line above the X-axis makes with the positive direction of the X-axis is:

1. Perpendicular line

2. Parallel line

3. Slope of a line

4. Concurrent line

Q. Tangent to the ellipse x^2 + 4 y^2=1 makes angle 60 degrees with the focal distance of the point of contact , then the square of the slope of the tangent is (A)1 (B)2 (C)3 (D)none of these

Q. $\mathrm{The}\mathrm{value}\left(s\right)\mathrm{of}\lambda \mathrm{for}\mathrm{which}\mathrm{the}\mathrm{line\; y=2x+\lambda }$touches the hyperbola $16x^2-9y^2=144$is/are:

1. $\sqrt{5}$
2. -$\sqrt{5}$
3. $2\sqrt{5}$
4. $-2\sqrt{5}$

Q. 28. let alfa be the angle which is tangent to the perabola yy=4ax makes its axis , the distance between the tangent and a perallel normal will be

Q. Show that the sum of intercepts of tangent to curve x+y =a on the coordinate axes is constant

Q. The ellipse $4x^2+9y^2=36$ and the hyperbola $4x^2-y^2=4$ have the same food and they intersect at right angles then the equation of the circle through the points of intersection of two conics is

1. $x^2+y^2=5$
2. $x^2+y^2=25$
3. $\sqrt{5}(x^2+y^2)+3x+4y=0$
4. $\sqrt{5}(x^2+y^2)-3x-4y=0$

Q. 78.The eq of the parabola whose focus is the point(00) and the tangent ay the vertix is x-y+1=0 is

Q. If $A\left(\theta \right)$ and $B\left(\varphi \right)$ are the parametric ends of a chord of hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ which passes through $(4,0)$, then the value of $\tan \frac{\theta }{2}\cdot \tan \frac{\varphi }{2}$is

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

Q. The equation of common tangent to the curves y^2=8x and xy=-1 is

Q. Tangent to the ellipse x^2 + 4 y^2=1 makes angle 60 degrees with the focal distance of the point of contact , then the square of the slope of the tangent is (A)1 (B)2 (C)3 (D)none of these

Q.

The slope of a line is double of the slope of another line. If tangent of the angle between them is, find the slopes of he lines.

Q. The slope of a line is double of the slope of another line. If tangents of the angle between them is $\frac{1}{3}$, find the slopes of the other line.

Q. { Q2. The locus of the foot of the perpendicular drawn from the centre of the ellipse }x^2+3y^2=6} on any tangent to it is

Q. Find the slope of the tangent to the curve f (x) = 2x⁶ + x⁴ − 1 at x = 1.

Q. Deduce the equation of the tangent of a parabola y²=4ax ( y=mx+a/m)

Q. The tangent to the parabola y²=-bx meets the parabola y²= 4ax in P and Q then mid point of PQ lies on the curve is

Q. The value of m for which y = mx + 6 is a tangent to the hyperbola $\frac{x^2}{100}-\frac{y^2}{49}=1$ is .

1. $\frac{\sqrt{17}}{10}$
2. $\frac{\sqrt{17}}{20}$
3. $\sqrt{\frac{17}{10}}$
4. $\sqrt{\frac{17}{20}}$