Tangents to ellipse

Find the magnitude of the angle at which the ellipse x² + 5 y² = 5 is visible from the point P[5, 1] .

Correct result:

A =  26.5651 °

Solution:

$$\begin{gathered} x^2+5y^2=5 \\ {P}_0=5;P_1=1 \\ {x}^2\mathrm{/}5+(y\mathrm{/}1{)}^2=1 \\ a=\sqrt{5}=\sqrt{5}\doteq 2.2361 \\ b=1 \\ {x}_0=0 \\ {y}_0=0 \\ {p}_1,p_2:y=k_x+q \\ {b}^2-q^2+a^2\cdot k^2=0 \\ 1-q^2+5\cdot k^2=0 \\ {P}_1=k\cdot P_0+q \\ 1-(P_1-k\ast P_0{)}^2+5\ast k^2=0 \\ 1-(1-k\cdot 5{)}^2+5\cdot k^2=0 \\ -20k^2+10k=0 \\ 20k^2-10k=0 \\ a=20;b=-10;c=0 \\ D=b^2-4ac=10^2-4\cdot 20\cdot 0=100 \\ D>0 \\ {k}_{1,2}={\displaystyle \frac{-b\pm \sqrt{D}}{2a}}={\displaystyle \frac{10\pm \sqrt{100}}{40}} \\ {k}_{1,2}={\displaystyle \frac{10\pm 10}{40}} \\ {k}_{1,2}=0.25\pm 0.25 \\ {k}_1=0.5 \\ {k}_2=0 \\ \text{ Factored form of the equation: } \\ 20(k-0.5)k=0 \\ {q}_1=P_1-k_1\cdot P_0=1-0.5\cdot 5=-{\displaystyle \frac{3}{2}}=-1.5 \\ {q}_2=P_1-k_2\cdot P_0=1-0\cdot 5=1 \\ {\phi }_1={\displaystyle \frac{180^{\circ }}{\pi }}\cdot \arctan (k_1)={\displaystyle \frac{180^{\circ }}{\pi }}\cdot \arctan (0.5)\doteq 26.5651^{\circ } \\ {\phi }_2={\displaystyle \frac{180^{\circ }}{\pi }}\cdot \arctan (k_2)={\displaystyle \frac{180^{\circ }}{\pi }}\cdot \arctan (0)=0^{\circ } \\ A=\mathrm{\mid }{\phi }_1-{\phi }_2\mathrm{\mid }=\mathrm{\mid }26.5651-0\mathrm{\mid }=26.5651^{\circ }=26^{\circ }33^{\mathrm{\prime }}54\mathrm{"} \end{gathered}$$

Our quadratic equation calculator calculates it.

Try another example

Showing 0 comments:

Next similar math problems:

• Find the 15
  Find the tangent line of the ellipse 9 x² + 16 y² = 144 that has the slope k = -1
• On line
  On line p: x = 4 + t, y = 3 + 2t, t is R, find point C, which has the same distance from points A [1,2] and B [-1,0].
• Ellipse
  Ellipse is expressed by equation 9x² + 25y² - 54x - 100y - 44 = 0. Find the length of primary and secondary axes, eccentricity, and coordinates of the center of the ellipse.
• Find the 5
  Find the equation of the circle with center at (1,20), which touches the line 8x+5y-19=0
• Equation of circle 2
  Find the equation of a circle which touches the axis of y at a distance 4 from the origin and cuts off an intercept of length 6 on the axis x.
• Sphere equation
  Obtain the equation of sphere its centre on the line 3x+2z=0=4x-5y and passes through the points (0,-2,-4) and (2,-1,1).
• Power line pole
  From point A, the power line pole is seen at an angle of 18 degrees. From point B to which we get when going from point A 30m away from the column at an angle of 10 degrees. Find the height of the power pole.
• Right triangle from axes
  A line segment has its ends on the coordinate axes and forms with them a triangle of area equal to 36 square units. The segment passes through the point ( 5,2). What is the slope of the line segment?
• On a line
  On a line p : 3 x - 4 y - 3 = 0, determine the point C equidistant from points A[4, 4] and B[7, 1].
• Curve and line
  The equation of a curve C is y=2x² -8x+9 and the equation of a line L is x+ y=3 (1) Find the x co-ordinates of the points of intersection of L and C. (2) Show that one of these points is also the stationary point of C?
• Parametric form
  Calculate the distance of point A [2,1] from the line p: X = -1 + 3 t Y = 5-4 t Line p has a parametric form of the line equation. ..
• Angle
  A straight line p given by the equation ?. Calculate the size of angle in degrees between line p and y-axis.
• Center of line segment
  Calculate the distance of the point X [1,3] from the center of the line segment x = 2-6t, y = 1-4t ; t is .
• Prove
  Prove that k1 and k2 is the equations of two circles. Find the equation of the line that passes through the centers of these circles. k1: x²+y²+2x+4y+1=0 k2: x²+y²-8x+6y+9=0
• Ratio iso triangle
  The ratio of the sides of an isosceles triangle is 7:6:7 Find the base angle to the nearest answer correct to 3 significant figure.
• Center
  In the triangle ABC is point D[1,-2,6], which is the center of the |BC| and point G[8,1,-3], which is the center of gravity of the triangle. Find the coordinates of the vertex A[x,y,z].
• General line equations
  In all examples, write the GENERAL EQUATION OF a line that is given in some way. A) the line is given parametrically: x = - 4 + 2p, y = 2 - 3p B) the line is given by the slope form: y = 3x - 1 C) the line is given by two points: A [3; -3], B [-5; 2] D) t