Equation of Tangent in Parametric Form

Q. Let $S$ be the circle in the $xy$-plane defined by the equation $x^2+y^2=4$.

Let $P$ be a point on the circle $S$ with both coordinates being positive. Let the tangent to $S$ at $P$ intersect the coordinate axes at the points $M$ and $N$. Then, the mid-point of the line segment $MN$ must lie on the curve

1. $(x+y{)}^2=3xy$
2. $x^{2/3}+y^{2/3}=2^{4/3}$
3. $x^2+y^2=2xy$
4. $x^2+y^2=x^2{y}^2$

Q. Let $S$ be the circle in the $xy$-plane defined by the equation $x^2+y^2=4$.

Let $P$ be a point on the circle $S$ with both coordinates being positive. Let the tangent to $S$ at $P$ intersect the coordinate axes at the points $M$ and $N$. Then, the mid-point of the line segment $MN$ must lie on the curve

1. $(x+y{)}^2=3xy$
2. $x^{2/3}+y^{2/3}=2^{4/3}$
3. $x^2+y^2=2xy$
4. $x^2+y^2=x^2{y}^2$

Q. The locus of the point of intersection of tangents to the circle $x=a\cos \theta ,y=a\sin \theta$ at the points, whose parametric angles differ by $\frac{\pi }{2}$ is

1. a pair of straight lines
2. a straight line
3. a circle
4. None of these

Q. The straight line $x\cos \theta +y\sin \theta =2$, will touch the circle $x^2+y^2-2x=0$, if

1. $\theta =n\pi ,n\in Z$
2. $\theta =(2n+1)\pi ,n\in Z$
3. $\theta =2n\pi ,n\in Z$
4. None of these

Q.

Let RS be the diameter of the circle $x^2+y^2=1$, where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersects a line drawn through Q parallel to RS at point E. Then, the locus of E passes through the point(s)

1. $(\frac{1}{3},\frac{1}{\sqrt{3}})$

2. $(\frac{1}{4},\frac{1}{2})$

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

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

Q.

Let RS be the diameter of the circle $x^2+y^2=1$, where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersects a line drawn through Q parallel to RS at point E. Then, the locus of E passes through the point(s)

1. $(\frac{1}{3},\frac{1}{\sqrt{3}})$

2. $(\frac{1}{4},\frac{1}{2})$

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

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

Q.

Let RS be the diameter of the circle $x^2+y^2=1$, where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersects a line drawn through Q parallel to RS at point E. Then, the locus of E passes through the point(s)

1. $(\frac{1}{3},\frac{1}{\sqrt{3}})$

2. $(\frac{1}{4},\frac{1}{2})$

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

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

Q.

Let RS be the diameter of the circle $x^2+y^2=1$, where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersects a line drawn through Q parallel to RS at point E. Then, the locus of E passes through the point(s)

1. $(\frac{1}{3},\frac{1}{\sqrt{3}})$

2. $(\frac{1}{4},\frac{1}{2})$

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

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

Q. Let $S$ be the circle in the $xy$-plane defined by the equation $x^2+y^2=4$.

Let $P$ be a point on the circle $S$ with both coordinates being positive. Let the tangent to $S$ at $P$ intersect the coordinate axes at the points $M$ and $N$. Then, the mid-point of the line segment $MN$ must lie on the curve

1. $(x+y{)}^2=3xy$
2. $x^{2/3}+y^{2/3}=2^{4/3}$
3. $x^2+y^2=2xy$
4. $x^2+y^2=x^2{y}^2$

Q. The straight line $x\cos \theta +y\sin \theta =2$, will touch the circle $x^2+y^2-2x=0$, if

1. $\theta =n\pi ,n\in Z$
2. $\theta =(2n+1)\pi ,n\in Z$
3. $\theta =2n\pi ,n\in Z$
4. None of these