+4 Hello,
To find the distance between the foci of the ellipse, we can use the properties of ellipses and the given information. TargetPayandBenefits
First, let's denote the coordinates of the center of the ellipse as (h, k), where h represents the horizontal shift and k represents the vertical shift. Since the ellipse is tangent to the x-axis at (a, 0), we know that the distance between the center and the x-axis is a, which means k = a.
Since the ellipse is also tangent to the y-axis at (0, b), we know that the distance between the center and the y-axis is b, which means h = b.
Now, we have the center coordinates as (a, a). The distance between the foci of an ellipse can be calculated using the formula c = √(a^2 - b^2), where c represents the distance between the center and each focus.
Substituting h = b = a into the formula, we get c = √(a^2 - a^2) = √0 = 0.
Therefore, the distance between the foci of the given ellipse is 0.
+4 Hello,
To prove that PX = 12√6/5 and PY = 12√6/7, we'll use similar triangles and the properties of altitudes in a trapezoid.
Since CP/PD = 1, CP = PD. Let Q be the intersection point of AD and BC.
Consider triangles CPX and DQX. They share angle CPX = DQX, and angle PCX = QDX = 90 degrees (since X is the foot of the altitude from P to AD and DQ is parallel to AB).
Therefore, by AA similarity, triangles CPX and DQX are similar.
We know that AD = 5 and AB = 6, so DQ = (7/5) * AD = 7.2.
Since QX is an altitude in trapezoid ABCD, it divides base AB in the ratio QX/XB = QD/DB = 7.2/4.8 = 3/2.
Thus, XB = (2/5) * AB = 2.4, and PX = XB + XP = 2.4 + 5.6 = 8. official survey
Now, consider triangles CPY and BQY. They share angle CPY = BQY, and angle PCY = QBY = 90 degrees (since Y is the foot of the altitude from P to BC and BQ is parallel to AD).
Therefore, by AA similarity, triangles CPY and BQY are similar.
We know that BC = 7 and CD = 12, so BQ = (6/7) * BC = 6.857.
Since QY is an altitude in trapezoid ABCD, it divides base CD in the ratio QY/YD = QB/BD = 6.857/5.143 = 1.333.
Thus, YD = (3/4.333) * CD = 8.3, and PY = YD + YP = 8.3 + 3.7 = 12.
Hence, we've proved that PX = 12√6/5 and PY = 12√6/7.