# Parametric Form of Normal : Ellipse

## Trending Questions

**Q.**If the normal at the point P(θ) to the ellipse x214+y25=1 intersects it again at the point Q(2θ), then

- sinθ=34
- cosθ=34
- cosθ=−23
- sinθ=−23

**Q.**

The normal to the curve $x=a\left(\mathrm{cos}\left(\theta \right)+\mathrm{sin}\left(\theta \right)\right),y=a\left(\mathrm{sin}\left(\theta \right)-\theta \mathrm{cos}\left(\theta \right)\right)$ at any point $\theta $ is such that

It is a constant distance from the origin

It passes through $\left(a\frac{\pi}{2},-a\right)$

It makes $\frac{\pi}{2}-\theta $ with the $x$-axis

It passes through the origin

**Q.**The eccentricity of an ellipse whose centre is at the origin is 12. If one of its directrices is x=−4, then the equation of the normal to it at (1, 32) is:

- 2y−x=2
- 4x−2y=1
- 4x+2y=7
- x+2y=4

**Q.**If CF be the perpendicular from the centre C of the ellipse x212+y28=1, on the tangent at any point P and G is the point where the normal at P meets the major axis, then the value of CF⋅PG=

**Q.**Any ordinate MP of the ellipse x225+y29=1 meets the auxiliary circle at Q, then locus of the point of intersection of normals at P and Q to the respective curves is

- x2+y2=8
- x2+y2=16
- x2+y2=34
- x2+y2=64

**Q.**The normal at a point P on the ellipse x2+4y2=16 meets the x− axis at Q. If M is the midpoint of the line segment PQ, then the locus of M intersects the latus rectum of the given ellipse at the points

- (±3√52, ±27)
- (±2√3, ±4√37)
- (±3√52, ±√197)
- (±2√3, ±17)

**Q.**The number of lines that can be drawn through the point (4, −5) at a distance 12 from the point (−2, 3) is:

- 0
- Infinite
- 1
- 2

**Q.**If normal at a variable point P on the ellipse x2a2+y2b2=1 of eccentricity e meets the axes of the ellipse at Q and R, then the locus of the midpoint of QR is a conic with an eccentricity e′ such that

- e′=e
- e′ is independent of e
- e′=e2
- e′=2e

**Q.**

Let $a$ and $b$ be positive real numbers such that $a>1$ and $b<a$. Let $P$ be a point in the first quadrant that lies on the hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$. Suppose the tangent to the hyperbola at $P$ passes through the point $\left(1,0\right)$, and suppose the normal to the hyperbola at $P$ cuts off equal intercepts on the coordinate axes. Let $\u2206$ denote the area of the triangle formed by the tangent at $P$, the normal at $P$ and the x-axis. If $e$ denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?

$1<e<\sqrt{2}$

$\sqrt{2}<e<2$

$\u2206={a}^{4}$

$\u2206={b}^{4}$

**Q.**If β is one of the angles between the normals to the ellipse, x2+3y2=9 at the points (3cosθ, √3sinθ) and (−3sinθ, √3cosθ); θ∈(0, π2); then 2cotβsin2θ is equal to

- 2√3
- 1√3
- √34
- √2

**Q.**The area of the rectangle (in sq. units) formed by the perpendiculars drawn from the centre of the ellipse having major axis and minor axis lengths as 2a and 2b units respectively to the tangent and normal at a point whose eccentric angle is π4 is

- (a2−b2)aba2+b2
- (a2+b2)aba2−b2
- (a2−b2)ab(a2+b2)
- (a2+b2)(a2−b2)ab

**Q.**Let x=4 be a directrix to an ellipse whose centre is at the origin and its eccentricity is 12. If P(1, β), β>0 is a point on this ellipse, then the equation of the normal to it at P is

- 8x−2y=5
- 4x−2y=1
- 7x−4y=1
- 4x−3y=2

**Q.**If the tangent drawn at point P(t2, 2t) on the parabola y2=4x is same as the normal drawn at point Q(√5cosθ, 2sinθ) on the ellipse 4x2+5y2=20, then

- Q≡(−1, 4√5)
- Q≡(−1, −4√5)
- P≡(15, −2√5)
- P≡(15, 2√5)

**Q.**If the tangent drawn at point (t2, 2t) on the parabola y2=4x is the same as the normal drawn at point (√5cosθ, 2sinθ) on the ellipse 4x2+5y2=20, then

- θ=cos−1(−1√5)
- θ=cos−1(1√5)
- t=−2√5
- t=−1√5

**Q.**P is the point on the ellipse x216+y29=1 and Q is the corresponding point on the auxilliary circle of the ellipse. If the line joining centre C to Q meets the normal at P with repsect to the given ellipse at K, then the value of CK is

**Q.**

The length of the normal to the curve $x=a\left(\theta +\mathrm{sin}\theta \right)$, $y=a\left(1-\mathrm{cos}\theta \right)$ at $\theta =\frac{\mathrm{\pi}}{2}$ is

$2a$

$\frac{a}{2}$

$\frac{a}{\sqrt{2}}$

$\sqrt{2}a$

**Q.**A line lx+my=n is normal to the ellipse x2a2+y2b2=1 under the condition n2(a2−b2)2(a2l2+b2m2)=k, then 7k=

**Q.**If the normal at θ to the ellipse 5x2+14y2=70, meets the curve again at 2θ, then the value of |3cosθ| is

**Q.**Which of the following is/are true ?

- There are infinite positive integral values of a for which (13x−1)2+(13y−2)2=(5x+12y−1a)2 represents an ellipse
- The minimum distance of a point (1, 2) from the ellipse 4x2+9y2+8x−36y+4=0 is 1 unit
- If from a point P(0, α) two normals other than axes are drawn to the ellipse x225+y216=1, then |α|<94
- If the length of latus rectum of an ellipse is one-third of its major axis, then its eccentricity is equal to 1√3

**Q.**

Describe parametric equation of a circle

**Q.**If Ax+By=5 is a normal at a point P on the ellipse x29+y24=1 whose eccentric angle is π4, then the value of (A+B)2 is

**Q.**Sum of distance's from the x−axis to the point(s) on the ellipse x29+y24=1, where the normal is parallel to the line 2x+y=1, is k5 unit, then k=

**Q.**If the latus rectum of the ellipse with axis along 𝑥−axis and centre at origin is 10, distance between foci = length of minor axis, then the equation of the ellipse is ––––––––

**Q.**

The normal of the curve $x=a\left(\mathrm{cos}\theta +\theta \mathrm{sin}\theta \right)$, $y=a\left(\mathrm{sin}\theta -\theta \mathrm{cos}\theta \right)$ at any $\theta $ is such that

It makes a constant angle with the $x$-axis

It passes through the origin

It is at a constant distance from the origin

None of the above

**Q.**The line lx + my + n = 0 will be a normal to the hyperbola b2x2−a2y2=a2b2, if

- a2l2+b2m2=(a2+b2)2n2

- a2l2+b2m2=(a2−b2)2n2

- a2l2+b2m2=(a2+b2)2n
- None of these

**Q.**

If $\sqrt{1-{x}^{6}}+\sqrt{1-{y}^{6}}=a({x}^{3}-{y}^{3})$ and $\frac{dy}{dx}=f(x,y)\sqrt{\left(\frac{1-{y}^{6}}{1-{x}^{6}}\right)}$, then

$f(x,y)=\frac{y}{x}$

$f(x,y)=2\left(\frac{y}{x}\right)$

$f(x,y)=\frac{{y}^{2}}{{x}^{2}}$

$f(x,y)=\frac{{x}^{2}}{{y}^{2}}$

**Q.**If a tangent on ellipse at A(1, 1) intersects its directrix at B(7, –7) and S be the corresponding focus of ellipse and C(α, β) is the circumcentre of △SAB, then

- α+β=1
- α−β=7
- SC2=20.5
- SC2=25

**Q.**cos(sin−1(tan(cos−1(sin(tan−143))))) is equal to

- 35
- √74
- 34
- 45

**Q.**Circles C1 and C2 , of radii r and R respectively, touch each other as shown in the figure. The line A, which is parallel to the line joining the centres of C1 and C2 , is tangent to C1 at P and intersects C2 at A, B. If R2=2r2, then ∠AOB equals

- 2212∘
- 45∘
- 60∘
- 6712∘

**Q.**An ellipse whose axis is along x axis and centre at origin. If the distance between focii is equal to length of minor axis, then the eccentricity of the ellipse is

- 12
- 1√3
- 1√2
- √2