Parametric Form of Tangent: Ellipse

Q.

If the normal at a point P to the hyperbola meets the transverse axis at G, and the value of $\frac{SG}{SP}$ is 2 then the eccentricity of the hyperbola is (where S is the focus of the hyperbola)

Q. Locus of the middle point of the intercept on the line y=x+c made by the lines 2x+3y=5 and 2x+3y=8, c being a parameter, is (1) 2x+3y+13=0 (2) 4x+6y+13=0 (3) 4x+6y-13=0 (4) 2x+3y-13=0

Q. A tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ cuts the axes in M and N. Then the least length of MN is

1. 
2. 
3. a + b
4. a - b

Q. A tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ cuts the axes in M and N. Then the least length of MN is

1. a + b
2. a - b
3. $a^2+b^2$
4. $a^2-b^2$

Q. Angle subtended by common tangents intercepted between two ellipses $4(x-4{)}^2+25y^2=100$ and $4(x+1{)}^2+y^2=4$ at origin is

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

Q. $\mathrm{If}\mathrm{the}\mathrm{line}$ $3x+4y=\sqrt{7}\mathrm{touches}\mathrm{the}\mathrm{ellipse}3x^2+4y^2=1,$
$\mathrm{then}\mathrm{the}\mathrm{coordinates}\mathrm{of}\mathrm{the}\mathrm{point}\mathrm{of}\mathrm{contact}\mathrm{is}$ .

1. $(\frac{\sqrt{7}}{3},0)$
2. $(\frac{1}{\sqrt{7}},\frac{1}{\sqrt{7}})$
3. $\left(0,\frac{\sqrt{7}}{4}\right)$
4. undefined

Q. If the tangent at point P on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the major axis at T, C is the centre and PN is the perpendicular on major axis, then the value of CN.CT is

1. $a^2$
2. $b^2$
3. $2a^2$
4. $2b^2$

Q. If the tangent at $\theta$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the auxiliary circle at two points which subtend a right angle at the centre, then $e^2(2-{\cos }^2\theta )=$

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

Q.

Tangent is drawn to ellipse $\frac{x^2}{27}+y^2=1at$
$(3\sqrt{3}cos\theta ,sin\theta )$ $(where,\theta \in (0,\frac{\pi }{2}))$.
Then, the value of $\theta$ such that the sum of intercepts on axes made by this tangent is minimum, is

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

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

3. $\frac{\pi }{8}$

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

Q. 50. Find the equations to the straight lines which go through the origin and trisect the portion of the straight line 3x + y = 12 which is intercepted between the axes of coordinates.

Q. A line passes through (2, 2) is perpendicular to the line 3x + y = 3, Its y intercept is (A) 1 3 (B) 2 3 (C) 1 (D) 4 3

Q. If PS is the altitude of the triangle with vertices P(2, 2), Q(6, −1) and R(7, 3) then the equation of line passing through (1, -1) and parallel to PS is