Circle And System Of Circles Question 40
Question: If the tangent at a point $ P(x,y) $ of a curve is perpendicular to the line that joins origin with the point P, then the curve is
[MP PET 1998]
Options:
A) $ ( \frac{-7}{2},-4 ) $
B) $ ( \frac{-18}{5},\frac{-21}{5} ) $
C) (2,-7)
D) (-2, -5)
Show Answer
Answer:
Correct Answer: B
Solution:
Let point of contact be $ P(x_1,\ y_1) $ . This point lies on line $ x_1+2y_1=-12 $ …………. (i) Gradient of $ OP=m_1=\frac{y_1-1}{x_1+1} $ Gradient of $ x+2y+12=m_2=-\frac{1}{2} $ The two lines are perpendicular,
$ \therefore \ m_1m_2=-1 $
$ \Rightarrow ( \frac{y_1-1}{x_1+1} )( \frac{-1}{2} )=-1\Rightarrow y_1-1=2x_1+2 $
$ \Rightarrow 2x_1-y_1=-3 $ …………. (ii) On
Solving equation (i) and (ii), we get $ (x_1,\ y_1)=( \frac{-18}{5},\ \frac{-21}{5} ) $ .
BETA
