Analytic Geometry Solutions
38 curated analytic geometry problems on conics — ellipse, hyperbola, parabola — with complete solutions and scoring rubrics.
38 problems
Problem 1: Find the standard equation of ellipse
Given the ellipse , which passes through and . A line through intersects the ellipse at points . Let be the origin. (1) Find the standard equation of ellipse .
Problem 2: Find the standard equation of ellipse
Given the ellipse , which passes through , and its eccentricity is . (1) Find the standard equation of ellipse . (2) Let be the upper vertex of ellipse . A line
Problem 3: Find the equation of ellipse
Given ellipse with eccentricity , and minor-axis length . A line not passing through the right vertex intersects the ellipse at points . (1) Find the equation o
Problem 4: Find the standard equation of ellipse
As shown in the figure, ellipse has eccentricity . Line passes through and the upper vertex of the ellipse, and its slope is . (1) Find the standard equation of
Problem 5: Find the equation of parabola
For parabola , the directrix is . Line intersects the parabola at points . (1) Find the equation of parabola . (2) If , find . (3) If there exist two points on
Problem 6: Find the equation of ellipse
Ellipse has eccentricity . Its left and right vertices are , its left and right foci are , and its upper vertex is . The circumradius of is . (1) Find the equat
Problem 7: Determine the positional relationship between line and circle , and justify your conclusion
Given the circle and the line . (1) Determine the positional relationship between line and circle , and justify your conclusion. (2) If is any point on circle ,
Problem 8: Find the equation of
Given ellipse , with left and right foci . For any point on , the perimeter of is , and the minimum value of is . (1) Find the equation of . (2) Let . A line th
Problem 9: Find the slope of
The directrix of parabola meets the -axis at . A line through the focus intersects the parabola at , . Points , lie on the directrix such that and . The circle
Problem 10: Find the slope of
The directrix of parabola meets the -axis at point . A line through the focus intersects the parabola at , . Points , lie on the directrix satisfying and . The
Problem 11: Find
For ellipse , tangents from touch at . Slopes are for , and . Find .
Problem 12: Find the maximum value of
In triangle , , , and point satisfies . Find the maximum value of .
Problem 13: find
For , line meets at . If and , find .
Problem 14: find the eccentricity of
For ellipse , let be the left and right foci. Let be any point on not coinciding with a vertex, and let be the incenter of . Let be the origin. Denote the slope
Problem 15: find fixed point
Given , , lines through meet them at . If is constant, find fixed point .
Problem 16: Find the range of the eccentricity
For hyperbola ( ), suppose that for any chord of the hyperbola (i.e. the circle with diameter always passes through the origin ), and that the vector inequality
Problem 17: Determine which of the following statements are true: (i) , (ii) If , then (iii) (iv)…
For parabola , let its directrix intersect the -axis at . A line through focus intersects at , , and the midpoint of is . Through , draw the line perpendicular
Problem 18: Find the fixed point on the -axis through which line always passes
Given ellipse with left and right foci , the focus of parabola coincides with . Let be the intersection point of the ellipse and parabola in the first quadrant,
Problem 19: Find
Consider the ellipse . Point does not coincide with either focus of . Let and be the reflections of about the two foci of , respectively. Let be a point such th
Problem 20: Find the minimum value of
Let and be the foci of the ellipse . Point moves on the ellipse. Find the minimum value of
Problem 21: find the perimeter of
Given the ellipse . Let be the upper endpoint of its minor axis, and its foci. The eccentricity is . The line through perpendicular to meets the ellipse at and
Problem 22: Consider the ellipse with left/right foci and left/right vertices
Consider the ellipse with left/right foci and left/right vertices . Point moves on the ellipse. Which statement is false? A. The eccentricity is . B. The perime
Problem 23: find
Let the ellipse be The upper and lower endpoints of its minor axis are and . A line given by intersects the ellipse at points and . Let the slopes of lines and
Problem 24: find the slope of
For the ellipse , a line through the left focus meets the ellipse at and (with above the -axis). If find the slope of . A. B. C. D.
Problem 25: Compute A
For the ellipse , let be its right focus. Point moves on the line . From , draw the two tangents and to the ellipse, touching it at and . Compute A. 3 B. 2 C. 1
Problem 26: Find the locus equation of point
Let the circle have center . A line passes through and is not the -axis. The line intersects the circle at and . Through , draw the line parallel to , meeting a
Problem 27: Find the perimeter of the focal triangle
Let be the ellipse with foci and . Point is any point on . Find the perimeter of the focal triangle .
Problem 28: Find the minimum possible value of
Let be the ellipse with foci , . Let be any point on the circle , and be any point on the circle . Point moves on . Find the minimum possible value of .
Problem 29: Find the standard equation of the ellipse
A central ellipse has equation . It is known that the tangent line to the ellipse at is . Find the standard equation of the ellipse.
Problem 30: find
Let be the ellipse . The line intersects at two distinct points . Let be the -coordinates of . If , find .
Problem 31: Find the equations of its asymptotes
A hyperbola is centered at the origin with transverse axis along the -axis. Its real axis length is and its eccentricity is . Find the equations of its asymptot
Problem 32: find the area of
Let be the hyperbola with foci and . Point lies on the right branch of . If , find the area of .
Problem 33: find and the equation of
Consider the hyperbola . Its asymptotes are two straight lines through the origin. The circle intersects these two asymptotes at four points, forming a rectangl
Problem 34: Find the length
Let be the hyperbola with right focus . The vertical line passes through and intersects at two points . Find the length .
Problem 35: Find the locus of points in the plane that are equidistant from the point and the…
Find the locus of points in the plane that are equidistant from the point and the line .
Problem 36: Find the chord length
Let be the parabola with focus . The line passes through and has slope . It intersects at two points . Find the chord length .
Problem 37: Find the minimum distance from the point to the curve , and give the point(s) on…
Let be the parabola . Find the minimum distance from the point to the curve , and give the point(s) on where it is attained.
Problem 38: Find the slope of line
Let be the parabola . A chord of has midpoint . Find the slope of line .