The position of the point $(1, 3)$ with respect to the ellipse $4{x^2} + 9{y^2} - 16x - 54y + 61 = 0$
The position of the point $(1, 3)$ with respect to the ellipse $4{x^2} + 9{y^2} - 16x - 54y + 61 = 0$
- A
Outside the ellipse
- B
On the ellipse
- C
On the major axis
- D
On the minor axis
Similar Questions
Consider two straight lines, each of which is tangent to both the circle $x ^2+ y ^2=\frac{1}{2}$ and the parabola $y^2=4 x$. Let these lines intersect at the point $Q$. Consider the ellipse whose center is at the origin $O (0,0)$ and whose semi-major axis is $OQ$. If the length of the minor axis of this ellipse is $\sqrt{2}$, then which of the following statement($s$) is (are) $TRUE$?
$(A)$ For the ellipse, the eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latus rectum is $1$
$(B)$ For the ellipse, the eccentricity is $\frac{1}{2}$ and the length of the latus rectum is $\frac{1}{2}$
$(C)$ The area of the region bounded by the ellipse between the lines $x=\frac{1}{\sqrt{2}}$ and $x=1$ is $\frac{1}{4 \sqrt{2}}(\pi-2)$
$(D)$ The area of the region bounded by the ellipse between the lines $x=\frac{1}{\sqrt{2}}$ and $x=1$ is $\frac{1}{16}(\pi-2)$
Consider two straight lines, each of which is tangent to both the circle $x ^2+ y ^2=\frac{1}{2}$ and the parabola $y^2=4 x$. Let these lines intersect at the point $Q$. Consider the ellipse whose center is at the origin $O (0,0)$ and whose semi-major axis is $OQ$. If the length of the minor axis of this ellipse is $\sqrt{2}$, then which of the following statement($s$) is (are) $TRUE$?
$(A)$ For the ellipse, the eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latus rectum is $1$
$(B)$ For the ellipse, the eccentricity is $\frac{1}{2}$ and the length of the latus rectum is $\frac{1}{2}$
$(C)$ The area of the region bounded by the ellipse between the lines $x=\frac{1}{\sqrt{2}}$ and $x=1$ is $\frac{1}{4 \sqrt{2}}(\pi-2)$
$(D)$ The area of the region bounded by the ellipse between the lines $x=\frac{1}{\sqrt{2}}$ and $x=1$ is $\frac{1}{16}(\pi-2)$
- [IIT 2018]
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