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GradeEquation Of Tangent of an Ellipse

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Any ordinate \[NP\] of an ellipse meets the auxiliary circle in \[Q\]; prove that the locus of the intersection of the normals at \[P\] and \[Q\] is the circle \[{{x}^{2}}+{{y}^{2}}={{\left( a+b \right)}^{2}}\].

If any tangent to the ellipse $\dfrac{{{x}^{2}}}{{{a}^{2}}}+\dfrac{{{y}^{2}}}{{{b}^{2}}}=1$ intercepts equal length $'l'$ on the axes, then $l$ is equal to

$\left( A \right)\text{ }{{a}^{2}}+{{b}^{2}}$

$\left( B \right)\text{ }\sqrt{{{a}^{2}}+{{b}^{2}}}$

$\left( C \right)\text{ }{{\left( {{a}^{2}}+{{b}^{2}} \right)}^{2}}$

$\left( D \right)\text{None of these}$

$\left( A \right)\text{ }{{a}^{2}}+{{b}^{2}}$

$\left( B \right)\text{ }\sqrt{{{a}^{2}}+{{b}^{2}}}$

$\left( C \right)\text{ }{{\left( {{a}^{2}}+{{b}^{2}} \right)}^{2}}$

$\left( D \right)\text{None of these}$

A tangent to the ellipse \[{x^2} + 4{y^2} = 4\] meets the ellipse \[{x^2} + 2{y^2} = 6\] at P and Q the tangents at P and Q of the ellipse \[{x^2} + 2{y^2} = 6\] is

A. \[{90^ \circ }\]

B. \[{60^ \circ }\]

C. \[{45^ \circ }\]

D. \[{30^ \circ }\]

A. \[{90^ \circ }\]

B. \[{60^ \circ }\]

C. \[{45^ \circ }\]

D. \[{30^ \circ }\]

If a tangent of slope m at a point of the ellipse $\dfrac{{{x}^{2}}}{{{a}^{2}}}+\dfrac{{{y}^{2}}}{{{b}^{2}}}=1$ passes through (2a, 0) and if āeā denotes the eccentricity of the ellipse, then:

(a)${{m}^{2}}+{{e}^{2}}=1$

(b) $2{{m}^{2}}+{{e}^{2}}=1$

(c) $3{{m}^{2}}+{{e}^{2}}=1$

(d)${{m}^{2}}+{{e}^{2}}-2=0$

(a)${{m}^{2}}+{{e}^{2}}=1$

(b) $2{{m}^{2}}+{{e}^{2}}=1$

(c) $3{{m}^{2}}+{{e}^{2}}=1$

(d)${{m}^{2}}+{{e}^{2}}-2=0$

Find the equations of tangent and normal to the ellipse \[2{x^2} + 3{y^2} = 11\] at the point whose ordinate is 1.

Find the point on the ellipse $4{{x}^{2}}+9{{y}^{2}}=1$ at which tangent is parallel to the line $8x=9y$.

Find the equation of tangent to the curve $4{{x}^{2}}+9{{y}^{2}}=36$ at the point $\left( 3\cos \theta ,2\sin \theta \right)$ where $\theta $ is the polar angle.

Write the equation of tangent at (1,1) on the curve \[2{{x}^{2}}+3{{y}^{2}}=5\].

Find equations of tangents drawn to the curve ${y^2}\, - \,2{x^2}\, - \,4y\, + \,8\, = \,0$ from the point $(\,1\,,\,2\,)$.

If tangents are drawn to the ellipse ${{x}^{2}}+2{{y}^{2}}=2$ at all points on the ellipse other than its four vertices, then the midpoints of the tangents intercepted between the coordinate axes lie on the curve:

(a) $\dfrac{{{x}^{2}}}{2}+\dfrac{{{y}^{2}}}{4}=1$,

(b) $\dfrac{{{x}^{2}}}{4}+\dfrac{{{y}^{2}}}{2}=1$,

(c) $\dfrac{1}{2{{x}^{2}}}+\dfrac{1}{4{{y}^{2}}}=1$,

(d) $\dfrac{1}{4{{x}^{2}}}+\dfrac{1}{2{{y}^{2}}}=1$.

(a) $\dfrac{{{x}^{2}}}{2}+\dfrac{{{y}^{2}}}{4}=1$,

(b) $\dfrac{{{x}^{2}}}{4}+\dfrac{{{y}^{2}}}{2}=1$,

(c) $\dfrac{1}{2{{x}^{2}}}+\dfrac{1}{4{{y}^{2}}}=1$,

(d) $\dfrac{1}{4{{x}^{2}}}+\dfrac{1}{2{{y}^{2}}}=1$.

The minimum area of a triangle formed by any tangent to the ellipse \[\dfrac{{{x^2}}}{{16}} + \dfrac{{{y^2}}}{{81}} = 1\] and the coordinate axes is

A. \[12\]

B.\[18\]

C.\[26\]

D.\[36\]

A. \[12\]

B.\[18\]

C.\[26\]

D.\[36\]

The equation of the tangent to the ellipse \[4{{x}^{2}}+3{{y}^{2}}=5\] which is parallel to the straight line \[y=3x+7\] is

A. \[2\sqrt{3}x-6\sqrt{3}y+\sqrt{155}=0\]

B. \[2\sqrt{3}x-2\sqrt{3}y-\sqrt{155}=0\]

C. \[6\sqrt{3}x-2\sqrt{3}y+\sqrt{155}=0\]

D. \[6\sqrt{3}x-2\sqrt{3}y-\sqrt{155}=0\]

A. \[2\sqrt{3}x-6\sqrt{3}y+\sqrt{155}=0\]

B. \[2\sqrt{3}x-2\sqrt{3}y-\sqrt{155}=0\]

C. \[6\sqrt{3}x-2\sqrt{3}y+\sqrt{155}=0\]

D. \[6\sqrt{3}x-2\sqrt{3}y-\sqrt{155}=0\]

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