Which of the following is/are true about the ellipse x2+4y2−2x−16y+13=0
A
The length of latus rectum of the ellipse is 1 unit
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
B
Distance between foci of the ellipse is 4√3 unit
No worries! We‘ve got your back. Try BYJU‘S free classes today!
C
Sum of the focal distances of a point P(x,y) on the ellipse is 4 unit
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
D
If the tangents drawn at the vertices of the ellipse meets line y=3 at the points P and Q then PQ always subtends a right angle at any of its foci
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
Open in App
Solution
The correct option is D If the tangents drawn at the vertices of the ellipse meets line y=3 at the points P and Q then PQ always subtends a right angle at any of its foci Given ellipse is x2+4y2−2x−16y+13=0 ⇒(x2−2x+1)+4(y2−4y+4)=4 ⇒(x−1)24+(y−2)21=1 ∴ Length of latus rectum =2b2a=2×12=1 unit
Also e=√1−b2a2=√1−14=√32
Distance between foci =2ae ⇒2ae=2×2×√32=2√3 unit
Sum of the focal distance =2a=4 unit
Tangents at the vertices are x−1=±a ⇒x−1=±2
or x=3,−1
Therefore, the line y=3 intersect these at point P(3,3)
and Q(−1,3).
Coordinates of focus are S(√3+1,2)
Slope of PS is 12−√3, slope of QS is 1−2−√3 ⇒ product of slopes =12−√3×1−2−√3=−1
It means PS⊥QS.