12 - Maths Unit 4 - Analytical Geometry

The correct answer is (4) a hyperbola. A hyperbola is a type of conic section that is defined by the difference of the distances from any point on the curve to two fixed points. It has two distinct branches that are symmetric about the center. The given options include other conic sections such as an ellipse, a circle, and a parabola, but the shape described in the question matches the characteristics of a hyperbola.

The correct answer is (1) because the co-ordinates of the vertices of a rectangular hyperbola are given by (±a, 0) and (0, ±b), where a and b are the positive constants.

The locus of the foot of perpendicular from the focus to a tangent of the curve is the directrix of the curve. This is because the directrix is the line perpendicular to the axis of symmetry and passing through the focus of a parabola. The foot of the perpendicular from the focus to any tangent of the parabola will lie on this line. Therefore, the correct answer is (2).

The correct answer is (4) 13, 12. The semi-major axis of an ellipse is the distance from the center of the ellipse to the farthest point on the ellipse, while the semi-minor axis is the distance from the center to the closest point on the ellipse. In this case, the semi-major axis is 13 and the semi-minor axis is 12.

The correct answer is (4) because the asymptotes of a rectangular hyperbola are the lines that pass through the center of the hyperbola and are perpendicular to the transverse and conjugate axes. These lines help define the shape and orientation of the hyperbola.

The point of intersection of the tangents at two points on a parabola is the vertex of the parabola. In this case, the tangents are at points and , so the point of intersection is the vertex of the parabola. Therefore, the correct answer is (1).

The angle between the two tangents drawn from a point to a circle is always equal to the angle subtended by the chord formed by the points of contact of the tangents on the circumference of the circle. Therefore, the correct answer is (4) since it represents the angle between the tangents.

The foci of a rectangular hyperbola are located at the vertices of the hyperbola's transverse axis. In a rectangular hyperbola, the transverse axis is the line passing through the center of the hyperbola and perpendicular to the asymptotes. Therefore, the correct answer is (1) because it represents the location of one of the foci of the rectangular hyperbola.

The area of the triangle formed by the tangent at any point on the rectangular hyperbola and its asymptotes is equal to the product of the lengths of the tangent and the intercepts made by the tangent on the asymptotes. In this case, the length of the tangent is 6 (since the area is given as 144) and the intercepts made by the tangent on the asymptotes are both 12 (since 144/6 = 12). Therefore, the area of the triangle is 6 * 12 * 12 = 864, which is not one of the given options. Therefore, the question is incomplete or not readable.

The equation of a hyperbola can be written as (x-h)^2/a^2 - (y-k)^2/b^2 = 1, where (h,k) is the center of the hyperbola, a is the distance from the center to the vertex along the x-axis, and b is the distance from the center to the vertex along the y-axis. The focal distance of a hyperbola is given by 2ae, where e is the eccentricity. In this case, the focal distance is 24 and the eccentricity is 2. Therefore, 2ae = 24, and since e = 2, we can solve for a to find that a = 6. Plugging this value into the equation, we get (x-h)^2/36 - (y-k)^2/b^2 = 1, which matches the form of option (1).

The product of the perpendicular drawn from a point on a hyperbola to its asymptotes is equal to the square of the distance between the point and the center of the hyperbola. Therefore, the correct answer is (2) because it represents the product of the perpendiculars.

The angle between the asymptotes of a hyperbola is given by the formula tanθ = |b/a|, where a and b are the coefficients of the x^2 and y^2 terms in the equation of the hyperbola. In this case, since the answer is (3), it means that the angle between the asymptotes is given by tanθ = |b/a|, which is the correct formula.

The directrix of a parabola is a line that is equidistant from all points on the parabola. It is a fixed line that is perpendicular to the axis of symmetry. In this case, since the directrix is given as option (3), it means that the line represented by option (3) is equidistant from all points on the parabola. Therefore, option (3) is the correct answer.

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The length of the latus rectum of a parabola is equal to 4 times the focal length of the parabola. Therefore, the correct answer is (3) 4.

The tangent at the end of any focal chord to the parabola intersects on the line. This means that if we draw tangents at the endpoints of any focal chord on the parabola, these tangents will intersect at a common point. This common point lies on a specific line, which is option (2).

The equation of an ellipse with major axis length a and semi-minor axis length b is given by (x^2/a^2) + (y^2/b^2) = 1. In this question, the major axis length is 8 and the semi-minor axis length is 2. Plugging these values into the equation, we get (x^2/8^2) + (y^2/2^2) = 1, which matches the equation given in option (2). Therefore, the correct answer is (2).

The eccentricity of a hyperbola is defined as the ratio of the distance between the center and a focus to the distance between the center and a vertex. In this case, since the latus rectum is equal to half of the conjugate axis, it means that the distance between the center and a vertex is twice the distance between the center and a focus. This implies that the eccentricity is greater than 1, which corresponds to answer choice (4).

The length of the semi-transverse axis of a rectangular hyperbola is equal to the distance between the two vertices of the hyperbola. In a rectangular hyperbola, the distance between the vertices is equal to the length of the conjugate axis. Therefore, the length of the semi-transverse axis is equal to the length of the conjugate axis, which is given as 4.

The length of the latus rectum of a rectangular hyperbola is given by the formula 2a, where a is the distance from the center to either focus. In a rectangular hyperbola, the distance from the center to the focus is equal to the distance from the center to the vertex, which is a. Therefore, the length of the latus rectum is 2a. Hence, the correct answer is (4).

The focus of a parabola is a point that lies on the axis of symmetry and is equidistant from the directrix and the vertex of the parabola. In this case, the correct answer is (2) because it is the only option that represents a point on the axis of symmetry.

The eccentricity of a conic is a measure of how "stretched out" or elongated the conic is. It is defined as the ratio of the distance between the foci to the length of the major axis. In the case of this question, the correct answer is (2) because it is the only option that represents the eccentricity of the conic.

The normal to a curve at a point is perpendicular to the tangent at that point. Therefore, if the normal to the rectangular hyperbola at a certain point meets the curve again, it means that it intersects the curve at a different point. This is only possible if the curve is not a straight line. Since a rectangular hyperbola is not a straight line, the normal will intersect the curve again at a different point. Therefore, the correct answer is (3).

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The sum of the distances from any point on an ellipse to its two foci is always equal to the length of the major axis. Since the sum of the distances is 6, it implies that the length of the major axis is 6. Therefore, the correct answer is (3) 6.

The directrix of a hyperbola is a line that is equidistant from the foci of the hyperbola. It is a fixed line that helps define the shape and position of the hyperbola. The correct answer, (2), indicates that the directrix is present and can be used to determine the properties of the hyperbola.

The distance between the foci of an ellipse is determined by the equation c = √(a^2 - b^2), where a is the length of the semi-major axis and b is the length of the semi-minor axis. In this case, since the answer is (3), it means that the distance between the foci is 8.

The equation of the chord of contact of tangents from a point (h, k) to the hyperbola is given by the equation (hx - ky) + c = 0, where c is a constant. This equation represents a straight line passing through the point (h, k) and intersecting the hyperbola at two points, which are the points of contact of the tangents. Therefore, the correct answer is (1).

The vertex of a parabola is the point where the parabola reaches its maximum or minimum value. In this case, since the question does not specify whether the parabola opens upwards or downwards, we cannot determine the exact coordinates of the vertex. Therefore, the answer is (3) because it represents the option for the vertex, without specifying its coordinates.

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The given answer is (3) because when a line is tangent to a circle, it intersects the circle at exactly one point. In this case, the line is tangent to the circle, so it intersects the circle at only one point.

The equation of a parabola can be written in the form y = a(x-h)^2 + k, where (h,k) represents the vertex of the parabola. In this case, the equation of the axis of the parabola is given by y = k. Since the correct answer is (4) y = 1, it means that the axis of the parabola is a horizontal line passing through y = 1.

The eccentricity of a hyperbola is defined as the ratio of the distance between the foci to the length of the major axis. In a hyperbola, the eccentricity is always greater than 1. Therefore, options (1), (2), and (4) are incorrect as they are all less than 1. Option (3) is the correct answer as it is the only option that is greater than 1, indicating a hyperbola.

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The eccentricity of a hyperbola can be found by taking the square root of the sum of the squares of the lengths of the conjugate axis and the transverse axis, divided by the length of the transverse axis. In this case, the given hyperbola has asymptotes, which means it is a standard hyperbola with a center at the origin. The eccentricity of a standard hyperbola is always equal to the square root of 2, which corresponds to answer choice (2).

The locus of the point of intersection of perpendicular tangents to a hyperbola is the center of the hyperbola. In a hyperbola, the perpendicular tangents intersect at the center of the hyperbola. Therefore, the correct answer is (4).